Smart thermostat with model predictive control and demand response integration

ABSTRACT

A system includes a plurality of thermostats corresponding to a plurality of HVAC systems that serve a plurality of spaces and a computing system communicable with the plurality of thermostats via a network. The computing system is configured to, for each space of the plurality of spaces, obtain a set of training data relating to thermal behavior of the space, identify a model of thermal behavior of the space based on the set of training data, perform a model predictive control process using the model of thermal behavior of the space to obtain a temperature setpoint for the space, and provide the temperature setpoint to the thermostat corresponding to the HVAC system serving the space. The plurality of thermostats are configured to control the plurality of HVAC systems in accordance with the temperature setpoints.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This application is a continuation-in-part of U.S. patent application Ser. No. 16/185,274, filed Nov. 9, 2018, which is a continuation of U.S. patent application Ser. No. 15/625,830 filed Jun. 16, 2017, which claims the benefit of and priority to U.S. Provisional Patent Application No. 62/491,545 filed Apr. 28, 2017, all of which are incorporated by reference herein in their entireties.

BACKGROUND

The present disclosure relates generally to a smart thermostat and more particularly to a smart thermostat with model predictive control. Thermostats are often configured to monitor and control the temperature of a building zone or other space. For example, a thermostat can be mounted on a wall within the building zone and configured to measure the temperature of the building zone. Thermostats typically send commands to HVAC equipment (e.g., on/off commands, heating/cooling commands, etc.) to cause the HVAC equipment to affect the temperature of the building zone.

Conventional thermostats operate according to a fixed temperature setpoint schedule which defines the temperature setpoints for the thermostat at various times. The temperature setpoint schedule is typically set by a user via a local user interface on the thermostat. In many implementations, a fixed temperature setpoint schedule leads to suboptimal control of the HVAC equipment, which can increase the cost of heating/cooling the building zone. It would be desirable to automatically determine optimal temperature setpoints for a thermostat in order to take advantage of time-varying energy prices, zone heat transfer characteristics, and/or other factors that can affect the cost of heating/cooling the building zone.

SUMMARY

One implementation of the present disclosure is a system. The system includes a plurality of thermostats corresponding to a plurality of HVAC systems that serve a plurality of spaces and a computing system communicable with the plurality of thermostats via a network. The computing system is configured to, for each space of the plurality of spaces, obtain a set of training data relating to thermal behavior of the space, identify a model of thermal behavior of the space based on the set of training data, perform a model predictive control process using the model of thermal behavior of the space to obtain a temperature setpoint for the space, and provide the temperature setpoint to the thermostat corresponding to the HVAC system serving the space. The plurality of thermostats are configured to control the plurality of HVAC systems in accordance with the temperature setpoints.

In some embodiments, the computing system is configured to obtain the set of training data relating to thermal behavior of the space by executing a system identification experiment for the space. Executing the system identification experiment includes providing a series of temperature setpoint changes to the thermostat corresponding to the HVAC system serving the space. The computing system may be configured to cause a temperature of the space to deviate from a normal comfort range while executing the system identification experiment. The thermostats may be configured to prompt users for permission to perform the system identification experiments. The system identification experiment may include causing the thermostat to display an indication that the system identification experiment is ongoing. Executing the system identification experiment may include preventing a user from adjusting a current temperature setpoint for the space.

In some embodiments, the computing system is configured to receive data from a utility provider computing system and adjust the model predictive control process based on the data from the utility provider computing system. In some embodiments, the computing system is configured to identify the model of thermal behavior of the space by performing a system identification process to determine values for parameters of the model of thermal behavior.

Another implementation of the present disclosure is a method for managing temperature in a space. The method includes defining a first temperature range for a normal operating mode, defining a second temperature range for a peak savings mode, operating in the normal operating mode by performing an optimization process to determine temperature setpoints predicted to maintain a temperature of the space within the first temperature range, receiving an indication from a utility company of an occurrence of a peak period, and operating in the peak savings mode for the peak period by performing the optimization process to determine temperature setpoints predicted to maintain the temperature of the space within the second temperature range during the peak period.

In some embodiments, defining the second temperature range includes prompting a user to input a maximum and minimum temperature for the peak savings mode. In some embodiments, defining the second temperature range includes expanding the first temperature range. In some embodiments, defining the second temperature range includes expanding the first temperature range by an amount prescribed by the utility company.

In some embodiments, the method includes, in response to the indication from the utility company of the occurrence of the peak period, prompting a user to opt-out of applying the peak savings mode for the peak period. The method may also include, in response to the user opting-out of applying the peak savings mode for the peak period, applying the normal operating mode during the peak period by performing the optimization process to determine temperature setpoints predicted to maintain the temperature of the space within the first temperature range during the peak period.

In some embodiments, prompting the user to opt-out of applying the peak savings mode for the peak period includes providing an option to opt-out on a graphical user interface of a thermostat. In some embodiments, prompting the user to opt-out of applying the peak savings mode for the peak period includes providing an option to opt-out on a graphical user interface of a smartphone. In some embodiments, prompting the user to opt-out of applying the peak savings mode for the peak period includes providing an estimate of cost savings associated with applying the peak savings mode for the peak period.

In some embodiments, the method includes operating heating or cooling equipment to drive the temperature of the space toward the temperature setpoints. In some embodiments, the optimization process includes a model predictive control process configured to minimize at least one of energy usage or energy cost over a time horizon subject to a temperature constraint. The temperature constraint may be defined by the first temperature range in the normal operating mode and by the second temperature range in the peak savings mode.

Another implementation of the present disclosure is a thermostat for equipment that operate to affect a temperature of a space. The thermostat includes a processing circuit configured to prompt a user for permission to initiate a training experiment, execute the training experiment by controlling the equipment over a duration of the training experiment to induce a dynamic response of the temperature of the space while collecting training data relating to a thermal behavior of the space, and control the equipment in a model predictive control mode in accordance with an output of an optimization process. The optimization process is based on a thermal model for the space identified using the training data.

In some embodiments, the processing circuit is configured to prevent the user from adjusting a temperature setpoint for the space during execution of the training experiment. In some embodiments, controlling the equipment over the duration of the training experiment to induce a dynamic response of the temperature of the space comprises varying a temperature setpoint for the space away from a user-selected temperature. In some embodiments, the processing circuit is configured to cause an indication that the training experiment is ongoing to be displayed during execution of the training experiment.

Another implementation of the present disclosure is a thermostat for monitoring and controlling temperature of a building zone. The thermostat includes an equipment controller and a model predictive controller. The equipment controller is configured to drive the temperature of the building zone to an optimal temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. The model predictive controller is configured to determine the optimal temperature setpoint by generating a cost function that accounts for a cost operating the HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, and optimizing the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps.

In some embodiments, the model predictive controller is configured to determine the cost of operating the HVAC equipment during each of the plurality of time steps using a set of time-varying utility rates comprising a utility rate value for each time step. The time-varying utility rates may be received from a utility provider or predicted by the model predictive controller.

In some embodiments, the model predictive controller is configured to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory comprising a temperature setpoint for each of the plurality of time steps.

In some embodiments, the model predictive controller is configured to optimize the cost function subject to a constraint on the optimal temperature setpoints that limits the optimal temperature setpoints within a temperature setpoint range.

In some embodiments, the model predictive controller is configured to generate the predictive model by performing a system identification process. The system identification process may include modulating the temperature setpoint within a constrained temperature setpoint range, collecting a set of input-output data, and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint during each of a plurality of time steps during a learning period.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of at least one of heat transfer between air within the building zone and solid mass within the building zone, heat transfer between the building zone and the HVAC equipment, and an unmeasured heat load disturbance. In some embodiments, the model predictive controller is configured to predict a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period.

In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint.

In some embodiments, the model predictive controller is configured to predict the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period.

In some embodiments, the constraint on the predicted temperature of the building zone requires the model predictive controller to maintain the predicted temperature of the building zone within a first zone temperature range during a first time step of the optimization period and within a second zone temperature range, different from the first zone temperature range, during another time step of the optimization period subsequent to the first time step.

Another implementation of the present disclosure is a method performed by a thermostat for a building zone for monitoring and controlling temperature of the building zone. The method includes generating a cost function that accounts for a cost operating HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, optimizing the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps, and operating HVAC equipment to provide heating or cooling to the building zone to drive the temperature of the building zone to the optimal temperature setpoints.

In some embodiments, the method includes receiving a set of time-varying utility rates from a utility provider or predicting the time-varying utility rates. The set of time-varying utility rates may include a utility rate value for each time step. The method may include determining the cost of operating the HVAC equipment during each of the plurality of time steps using the set of time-varying utility rates.

In some embodiments, using the predictive model to predict the temperature of the building zone includes predicting the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory that includes a temperature setpoint for each of the plurality of time steps.

In some embodiments, optimizing the cost function includes optimizing the cost function subject to a constraint on the optimal temperature setpoints that limits the optimal temperature setpoints within a temperature setpoint range.

In some embodiments, the method includes generating the predictive model by performing a system identification process. The system identification process may include modulating the temperature setpoint within a constrained temperature setpoint range, collecting a set of input-output data, and fitting parameters of the predictive model to the set of input-output data. The input-output data include values of the temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint during each of a plurality of time steps during a learning period.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of at least one of heat transfer between air within the building zone and solid mass within the building zone, heat transfer between the building zone and the HVAC equipment, and an unmeasured heat load disturbance. In some embodiments, the method includes predicting a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period.

In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint.

In some embodiments, the method includes predicting the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period.

Another implementation of the present disclosure is a thermostat for monitoring and controlling temperature of a building zone. The thermostat includes an equipment controller and a model predictive controller. The equipment controller is configured to drive the temperature of the building zone to a zone temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. The model predictive controller is configured to determine the zone temperature setpoint by generating a cost function that accounts for a cost operating the HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, and performing an optimization of the cost function subject to a constraint on the predicted temperature of the building zone to determine a temperature setpoint trajectory including a temperature setpoint value for each of the plurality of time steps.

Another implementation of the present disclosure is a model predictive control system for monitoring and controlling temperature of a building zone. The model predictive control system includes a thermostat and a model predictive controller. The thermostat is configured to drive the temperature of the building zone to an optimal temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. The model predictive controller is configured to determine the optimal temperature setpoint and provide the optimal temperature setpoint to the thermostat via a communications network. The model predictive controller determines the optimal temperature setpoint by generating a cost function that accounts for a cost operating the HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, and optimizing the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps.

In some embodiments, the model predictive controller is configured to determine the cost of operating the HVAC equipment during each of the plurality of time steps using a set of time-varying utility rates that includes a utility rate value for each time step. The time-varying utility rates may be received from a utility provider or predicted by the model predictive controller.

In some embodiments, the model predictive controller is configured to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory comprising a temperature setpoint for each of the plurality of time steps.

In some embodiments, the model predictive controller is configured to optimize the cost function subject to a constraint on the optimal temperature setpoints that limits the optimal temperature setpoints within a temperature setpoint range.

In some embodiments, the model predictive controller is configured to generate the predictive model by performing a system identification process. The system identification process may include modulating the temperature setpoint within a constrained temperature setpoint range, collecting a set of input-output data, and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint during each of a plurality of time steps during a learning period.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of at least one of heat transfer between air within the building zone and solid mass within the building zone, heat transfer between the building zone and the HVAC equipment, and an unmeasured heat load disturbance. In some embodiments, the model predictive controller is configured to predict a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period.

In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint.

In some embodiments, the model predictive controller is configured to predict the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period.

In some embodiments, the constraint on the predicted temperature of the building zone requires the model predictive controller to maintain the predicted temperature of the building zone within a first zone temperature range during a first time step of the optimization period and within a second zone temperature range, different from the first zone temperature range, during another time step of the optimization period subsequent to the first time step.

Another implementation of the present disclosure is a method for monitoring and controlling temperature of a building zone. The method includes generating a cost function that accounts for a cost operating HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, optimizing the cost function at a model predictive controller subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps, and providing an optimal temperature setpoint from the model predictive controller to a thermostat for the building zone via a communications network. The method further includes, at the thermostat, driving the temperature of the building zone to the optimal temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone.

In some embodiments, the method includes receiving a set of time-varying utility rates from a utility provider or predicting the time-varying utility rates. The set of time-varying utility rates may include a utility rate value for each time step. The method may include determining the cost of operating the HVAC equipment during each of the plurality of time steps using the set of time-varying utility rates.

In some embodiments, using the predictive model to predict the temperature of the building zone includes predicting the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory that includes a temperature setpoint for each of the plurality of time steps.

In some embodiments, optimizing the cost function includes optimizing the cost function subject to a constraint on the optimal temperature setpoints that limits the optimal temperature setpoints within a temperature setpoint range.

In some embodiments, the method includes generating the predictive model by performing a system identification process. The system identification process may include modulating the temperature setpoint within a constrained temperature setpoint range, collecting a set of input-output data, and fitting parameters of the predictive model to the set of input-output data. The input-output data include values of the temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint during each of a plurality of time steps during a learning period.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of at least one of heat transfer between air within the building zone and solid mass within the building zone, heat transfer between the building zone and the HVAC equipment, and an unmeasured heat load disturbance. In some embodiments, the method includes predicting a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period.

In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint.

In some embodiments, the method includes predicting the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period.

Another implementation of the present disclosure is a model predictive control system for monitoring and controlling temperature of a building zone. The model predictive control system includes a thermostat and a model predictive controller. The thermostat is configured to drive the temperature of the building zone to a zone temperature setpoint by operating HVAC equipment to provide heating or cooling to the building zone. The model predictive controller is configured to determine the zone temperature setpoint and provide the zone temperature setpoint to the thermostat via a communications network. The model predictive controller determines the zone temperature setpoint by generating a cost function that accounts for a cost operating the HVAC equipment during each of a plurality of time steps in an optimization period, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps, and performing an optimization of cost function subject to a constraint on the predicted temperature of the building zone to determine the temperature setpoint trajectory.

Another implementation of the present disclosure is a model predictive controller for monitoring and controlling temperature of a building zone. The model predictive controller includes a system identifier and a predictive optimizer. The system identifier is configured to operate the model predictive controller in a system identification mode. Operating in the system identification mode includes performing a system identification process to automatically generate a predictive model based on heat transfer characteristics of the building zone. The predictive optimizer is configured to operate the model predictive controller in an operational mode. Operating in the operational mode includes using the predictive model to predict the temperature of the building zone. The model predictive controller is configured to automatically transition from the operational mode to the system identification mode in response to a determination that a prediction error of the predictive model exceeds a threshold value.

In some embodiments, operating in the operational mode includes optimizing a cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of a plurality of time steps in an optimization period. In some embodiments, the cost function accounts for a cost of operating HVAC equipment to provide heating or cooling to the building zone during each of the plurality of time steps in the optimization period.

In some embodiments, performing the system identification process includes collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of a temperature setpoint for the building zone and the temperature of the building zone during each of a plurality of time steps during a learning period.

In some embodiments, the set of input-output data includes a discrete HVAC staging trajectory for staged HVAC equipment. The HVAC staging trajectory may include a discrete HVAC equipment load at each of the plurality of time steps during the learning period. In some embodiments, performing the system identification process includes filtering the input-output data to generate a continuous HVAC equipment load signal from the discrete HVAC staging trajectory.

In some embodiments, collecting the set of input-output data includes modulating a temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint. In some embodiments, performing the system identification process comprises filtering the input-output data to remove oscillations in the temperature of the building zone around the temperature setpoint resulting from activating and deactivating staged HVAC equipment.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone.

In some embodiments, the model predictive controller includes a load/rate predictor configured to predict a value of an unmeasured heat load disturbance experienced by the building zone at each of a plurality of time steps in an optimization period. In some embodiments, the predictive model defines the temperature of the building zone as a function of the unmeasured heat load disturbance.

In some embodiments, the predictive model includes an HVAC load model that defines an amount of heating or cooling provided by HVAC equipment controlled by the model predictive controller as a function of the temperature of the building zone and a temperature setpoint for the building zone.

Another implementation of the present disclosure is a method for monitoring and controlling temperature of a building zone. The method includes operating a model predictive controller in a system identification mode. Operating in the system identification mode includes performing a system identification process to automatically generate a predictive model based on heat transfer characteristics of the building zone. The method further includes operating the model predictive controller in an operational mode. Operating in the operational mode includes using the predictive model to predict the temperature of the building zone. The method further includes automatically transitioning from the operational mode to the system identification mode in response to a determination that a prediction error of the predictive model exceeds a threshold value.

In some embodiments, operating in the operational mode includes optimizing a cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of a plurality of time steps in an optimization period. In some embodiments, the cost function accounts for a cost of operating HVAC equipment to provide heating or cooling to the building zone during each of the plurality of time steps in the optimization period.

In some embodiments, performing the system identification process includes collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of a temperature setpoint for the building zone and the temperature of the building zone during each of a plurality of time steps during a learning period.

In some embodiments, the set of input-output data includes a discrete HVAC staging trajectory for staged HVAC equipment. The HVAC staging trajectory may include a discrete HVAC equipment load at each of the plurality of time steps during the learning period. In some embodiments, performing the system identification process includes filtering the input-output data to generate a continuous HVAC equipment load signal from the discrete HVAC staging trajectory.

In some embodiments, collecting the set of input-output data comprises modulating a temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint. In some embodiments, performing the system identification process includes filtering the input-output data to remove oscillations in the temperature of the building zone around the temperature setpoint resulting from activating and deactivating staged HVAC equipment.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone.

In some embodiments, the method includes predicting a value of an unmeasured heat load disturbance experienced by the building zone at each of a plurality of time steps in an optimization period. In some embodiments, the predictive model defines the temperature of the building zone as a function of the unmeasured heat load disturbance.

In some embodiments, the predictive model includes an HVAC load model that defines an amount of heating or cooling provided by HVAC equipment controlled by the model predictive controller as a function of the temperature of the building zone and a temperature setpoint for the building zone.

Another implementation of the present disclosure is a model predictive controller for monitoring and controlling temperature of a building zone. The model predictive controller includes a state/disturbance estimator and a predictive optimizer. The state/disturbance estimator is configured to estimate an initial state of the building zone at a beginning of an optimization period. The state of the building zone includes a temperature of air within the building zone and a temperature of solid mass within the building zone. The predictive optimizer is configured to use a predictive model of the building zone to predict the state of the building zone at each of a plurality of time steps of the optimization period based on the estimated initial state of the building zone and a temperature setpoint trajectory comprising a temperature setpoint for each of the plurality of time steps. The predictive optimizer is configured to generate optimal temperature setpoints for each of the plurality of time steps by optimizing a cost function that accounts for a cost operating HVAC equipment during each of the plurality of time steps.

In some embodiments, the predictive optimizer is configured to generate the cost function and determine the cost operating the HVAC equipment during each time step using a set of time-varying utility rates that includes a utility rate value for each time step. The set of time-varying utility rates can be received from a utility provider or predicted by the model predictive controller.

In some embodiments, the predicted state of the building zone includes a predicted temperature of the building zone. The predictive optimizer may be configured to optimize the cost function subject to a constraint on the predicted temperature of the building zone.

In some embodiments, the model predictive controller includes a load/rate predictor configured to predict a value of an unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period. In some embodiments, the predictive model defines the temperature of the building zone as a function of the unmeasured heat load disturbance experienced by the building zone at each time step.

In some embodiments, the predictive model includes a mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. In some embodiments, the predictive model defines the temperature of the building zone as a function of heat transfer between the building zone and the HVAC equipment. In some embodiments, the predictive model includes an HVAC load model that defines an amount of heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint.

In some embodiments, the predictive optimizer is configured to predict the cost of operating the HVAC equipment as a function of an amount of heating or cooling provided by the HVAC equipment at each time step of the optimization period.

In some embodiments, the model predictive controller includes a system identifier configured to generate the predictive model by performing a system identification process. The system identification process may include collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and the temperature of the building zone during each of a plurality of time steps during a learning period.

In some embodiments, performing system identification process includes modulating the temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint.

Another implementation of the present disclosure is a method for monitoring and controlling temperature of a building zone. The method includes estimating an initial state of the building zone at a beginning of an optimization period. The state of the building zone includes a temperature of air within the building zone and a temperature of solid mass within the building zone. The method includes using a predictive model of the building zone to predict the state of the building zone at each of a plurality of time steps of the optimization period based on the estimated initial state of the building zone and a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps. The method includes generating optimal temperature setpoints for each of the plurality of time steps by optimizing a cost function that accounts for a cost operating HVAC equipment during each of the plurality of time steps.

In some embodiments, the method includes generating the cost function and determining the cost operating the HVAC equipment during each time step using a set of time-varying utility rates comprising a utility rate value for each time step. The set of time-varying utility rates can be received from a utility provider or predicted as part of the method.

In some embodiments, the predicted state of the building zone includes a predicted temperature of the building zone. The method may include optimizing the cost function subject to a constraint on the predicted temperature of the building zone.

In some embodiments, the method includes predicting a value of an unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period. In some embodiments, the predictive model defines the temperature of the building zone as a function of the unmeasured heat load disturbance experienced by the building zone at each time step.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. In some embodiments, the predictive model defines the temperature of the building zone as a function of heat transfer between the building zone and the HVAC equipment. In some embodiments, the predictive model includes an HVAC load model that defines an amount of heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint.

In some embodiments, the method includes predicting the cost of operating the HVAC equipment as a function of an amount of heating or cooling provided by the HVAC equipment at each time step of the optimization period.

In some embodiments, the method includes generating the predictive model by performing a system identification process. The system identification process includes collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and the temperature of the building zone during each of a plurality of time steps during a learning period.

Another implementation of the present disclosure is a model predictive controller for monitoring and controlling temperature of a building zone. The model predictive controller includes a state/disturbance estimator and a predictive optimizer. The state/disturbance estimator is configured to estimate an initial state of the building zone at a beginning of an optimization period. The state of the building zone includes a temperature of air within the building zone and a temperature of solid mass within the building zone. The predictive optimizer is configured to use a predictive model of the building zone to predict the state of the building zone at each of a plurality of time steps of the optimization period based on the estimated initial state of the building zone and a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps. The predictive optimizer is configured to generate temperature setpoints for each of the plurality of time steps by optimizing a cost function that accounts for a cost operating HVAC equipment during each of the plurality of time steps.

Another implementation of the present disclosure is a thermostat for monitoring and controlling temperature of a building zone. The thermostat includes a load/rate predictor and a predictive optimizer. The load/rate predictor is configured to predict a price of one or more resources consumed by HVAC equipment to generate heating or cooling for the building zone at each of a plurality of time steps of an optimization period. The predictive optimizer is configured to generate a cost function that accounts for a cost of operating the HVAC equipment during each time step as a function of the predicted prices at each time step, use a predictive model to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps, and optimize the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps.

In some embodiments, the thermostat includes an equipment controller configured to drive the temperature of the building zone to an optimal temperature setpoint by operating the HVAC equipment to provide heating or cooling to the building zone.

In some embodiments, the predictive model defines the temperature of the building zone as a function of an unmeasured heat load disturbance. In some embodiments, the load/rate predictor is configured to predict a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period as a function of at least one of day type, time of day, building occupancy, outdoor air temperature, and weather forecasts.

In some embodiments, the thermostat includes a system identifier configured to generate the predictive model by performing a system identification process. The system identification process includes collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and the temperature of the building zone during each of a plurality of time steps during a learning period.

In some embodiments, the system identification process includes modulating the temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. In some embodiments, the predictive model defines the temperature of the building zone as a function of heat transfer between the building zone and the HVAC equipment. In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint.

In some embodiments, the predictive optimizer is configured to predict the cost of operating the HVAC equipment as a function of an amount of the heating or cooling provided by the HVAC equipment at each time step of the optimization period.

Another implementation of the present disclosure is a method performed by a thermostat for a building zone for monitoring and controlling temperature of the building zone. The method includes predicting a price of one or more resources consumed by HVAC equipment to generate heating or cooling for the building zone at each of a plurality of time steps of an optimization period, generating a cost function that accounts for a cost of operating the HVAC equipment during each time step as a function of the predicted prices at each time step, using a predictive model to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps, and optimizing the cost function subject to a constraint on the predicted temperature of the building zone to determine optimal temperature setpoints for each of the plurality of time steps.

In some embodiments, the method includes driving the temperature of the building zone to an optimal temperature setpoint by operating the HVAC equipment to provide heating or cooling to the building zone.

In some embodiments, the predictive model defines the temperature of the building zone as a function of an unmeasured heat load disturbance. In some embodiments, the method includes predicting a value of the unmeasured heat load disturbance experienced by the building zone at each of the plurality of time steps in the optimization period as a function of at least one of day type, time of day, building occupancy, outdoor air temperature, and weather forecasts.

In some embodiments, the method includes generating the predictive model by performing a system identification process. The system identification process may include collecting a set of input-output data and fitting parameters of the predictive model to the set of input-output data. The input-output data may include values of the temperature setpoint and the temperature of the building zone during each of a plurality of time steps during a learning period.

In some embodiments, performing the system identification process includes modulating the temperature setpoint within a constrained temperature setpoint range and recording values of the modulated temperature setpoint and values of the temperature of the building zone that result from modulating the temperature setpoint.

In some embodiments, the predictive model includes a thermal mass storage model that defines the temperature of the building zone as a function of heat transfer between air within the building zone and solid mass within the building zone. In some embodiments, the predictive model defines the temperature of the building zone as a function of heat transfer between the building zone and the HVAC equipment. In some embodiments, the predictive model includes an HVAC load model that defines the heating or cooling provided by the HVAC equipment as a function of the temperature of the building zone and the temperature setpoint.

Another implementation of the present disclosure is a thermostat for monitoring and controlling temperature of a building zone. The thermostat includes a load/rate predictor and a predictive optimizer. The load/rate predictor is configured to predict a price of one or more resources consumed by HVAC equipment to generate heating or cooling for the building zone at each of a plurality of time steps of an optimization period. The predictive optimizer is configured to generate a cost function that accounts for a cost of operating the HVAC equipment during each time step as a function of the predicted prices at each time step, use a predictive model to predict the temperature of the building zone during each of the plurality of time steps as a function of a temperature setpoint trajectory including a temperature setpoint for each of the plurality of time steps, and perform an optimization of the cost function subject to a constraint on the predicted temperature of the building zone to determine temperature setpoints for each of the plurality of time steps.

Those skilled in the art will appreciate that the summary is illustrative only and is not intended to be in any way limiting. Other aspects, inventive features, and advantages of the devices and/or processes described herein, as defined solely by the claims, will become apparent in the detailed description set forth herein and taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a drawing of a smart thermostat having a landscape aspect ratio, according to some embodiments.

FIG. 2 is a drawing of a smart thermostat having a portrait aspect ratio, according to some embodiments.

FIG. 3 is a block diagram of a model predictive control system including a model predictive controller and a smart thermostat which communicates with the model predictive controller via a network, according to some embodiments.

FIG. 4 is a block diagram of a model predictive control system including a smart thermostat that includes a model predictive controller and an equipment controller, according to some embodiments.

FIG. 5 is a block diagram illustrating the model predictive controller of FIGS. 3-4 in greater detail, according to some embodiments.

FIG. 6 is a block diagram illustrating various types of predictive models, according to some embodiments.

FIG. 7 is a graph of a temperature excitation signal over a one-day period, according to some embodiments.

FIG. 8 is a thermal circuit diagram that models the heat transfer characteristics of a building zone, according to some embodiments.

FIG. 9 is a flowchart of a process which can be performed by the model predictive controller of FIG. 5 to identify parameters of a predictive model, according to some embodiments.

FIG. 10 is a graph of an unfiltered temperature signal and an unfiltered sensible load signal which can be received as feedback from a controlled HVAC system, according to some embodiments.

FIG. 11 is a graph of a filtered temperature signal and a filtered sensible load signal which can be generated by filtering the signals shown in FIG. 10, according to some embodiments.

FIG. 12 is a flowchart of a process for operating the model predictive controller of FIG. 5 in a system identification mode and in an operational mode, according to some embodiments.

FIG. 13 is a flowchart of a process which can be performed by the model predictive controller of FIG. 5 to determine optimal temperature setpoints when operating in the operational mode, according to some embodiments.

FIG. 14 is a flowchart of a process which can be performed by the model predictive controller of FIG. 5 to generate a predictive model and use the predictive model to determine optimal temperature setpoints for a building zone, according to some embodiments.

FIG. 15 is a graph comparing the energy consumption of the model predictive control systems of FIGS. 3-4 with a baseline system that does not use model predictive control, according to some embodiments.

FIG. 16 is a graph of time-varying electricity prices illustrating different electricity prices at different times, according to some embodiments.

FIG. 17 is a graph comparing the building mass temperature in the model predictive control systems of FIGS. 3-4 with a baseline system that does not use model predictive control, according to some embodiments.

FIG. 18 is a graph comparing the zone air temperature in the model predictive control systems of FIGS. 3-4 with a baseline system that does not use model predictive control, according to some embodiments.

FIG. 19 is a block diagram of the computing architecture for a network of smart thermostats with model predictive control and demand response integration, according to some embodiments.

FIG. 20 is a flowchart of a process for obtaining a system model for a space served by a smart thermostat of FIG. 19, according to some embodiments.

FIG. 21 is a view of a graphical user interface displayed on a smart thermostat for prompting a user to opt-in to an energy aid mode, according to some embodiments.

FIG. 22 is a view of a graphical user interface displayed on a smart thermostat for prompting a user for permission to initiate a system identification experiment, according to some embodiments.

FIG. 23 is a view of a graphical user interface displayed on a smart thermostat for providing an indication that a system identification experiment is ongoing, according to some embodiments.

FIG. 24 is a view of a graphical user interface displayed on a smart thermostat for providing information to a user regarding an ongoing system identification experiment, according to some embodiments.

FIG. 25 is a flowchart of a process for activate a peak savings mode for the smart thermostat, according to some embodiments.

FIG. 26 is a view of a graphical user interface displayed on a smart thermostat for prompting a user to opt-in to a peak savings mode, according to some embodiments.

FIG. 27 is a view of a graphical user interface displayed on a smart thermostat for prompting a user to set temperature bounds for a peak savings mode, according to some embodiments.

FIG. 28 is a flowchart of a process for providing a peak savings mode for the smart thermostat, according to some embodiments.

FIG. 29 is a view of a graphical user interface displayed on a smart thermostat for informing a user of an upcoming peak savings period and allowing the user to opt-out for the upcoming period, according to some embodiments.

FIG. 30 is a view of a graphical user interface displayed on a smart thermostat for informing a user of a current peak savings period and allowing the user to opt-out for the current period, according to some embodiments.

FIG. 31 is a view of a graphical user interface that includes a graph showing energy savings over time, according to some embodiments.

DETAILED DESCRIPTION

Overview

Referring generally to the FIGURES, a smart thermostat with model predictive control and components thereof are shown according, to some embodiments. The smart thermostat can be configured to monitor and control one or more environmental conditions of a building zone (e.g., temperature, humidity, air quality, etc.). In some embodiments, the smart thermostat is mounted on a wall within the building zone and configured to measure the temperature, humidity, and/or other environmental conditions of the building zone. The smart thermostat can be configured to communicate with HVAC equipment that operates to affect the measured environmental conditions.

In some embodiments, the smart thermostat includes a model predictive controller and an equipment controller. In other embodiments, the model predictive controller is separate from the smart thermostat and communicates with the smart thermostat via a communications network (e.g., the Internet, a building network, etc.). The model predictive controller can be configured to determine optimal temperature setpoints T_(sp) for the building zone for each of a plurality of time steps in an optimization period. The equipment controller can be configured to receive the optimal temperature setpoints T_(sp) from the model predictive controller and can operate HVAC equipment to drive the temperature of the building zone to the optimal temperature setpoints.

To determine the optimal temperature setpoints T_(sp), the model predictive controller can optimize an objective function (i.e., a cost function) that accounts for the cost of operating the HVAC equipment over the duration of the optimization period. The costs of operating the HVAC equipment can include, for example, the costs of resources consumed by the HVAC equipment during operation (e.g., electricity, natural gas, water, etc.), demand charges imposed by an electric utility, peak load contribution charges, equipment degradation/replacement costs, and/or other costs associated with the operation of the HVAC equipment. The optimization performed by the model predictive controller is described in greater detail below.

The model predictive controller can optimize the objective function subject to a set of constraints. The constraints may include temperature constraints (e.g., a maximum temperature limit and a minimum temperature limit for the building zone), equipment capacity constraints, load change constraints, thermal mass storage constraints, HVAC load constraints, and/or other constraints that limit the operation of the HVAC equipment and/or describe the temperature evolution of the building zone. The model predictive controller can automatically generate the optimization constraints by performing a system identification process and generating predictive models that describe the controlled system (i.e., the HVAC equipment and the building zone).

In some embodiments, the model predictive controller generates a thermal mass storage model that describes the temperature of the air within the building zone T_(ia) as a function of several physical parameters and variables. The physical parameters in the thermal mass storage model may include, for example, the thermal capacitance C_(ia) of the air in the building zone, the thermal capacitance C_(m) of the solid mass within the building zone, the thermal resistance R_(mi) between the air within the building zone and the solid mass, the thermal resistance R_(oi) between the air within the building zone and the ambient environment, etc. The variables in the thermal mass storage model may include the temperature of air within the building zone T_(ia), the temperature of the solid mass within the building zone T_(m), the outside air temperature T_(oa), the amount of heating or cooling provided by the HVAC equipment {dot over (Q)}_(HVAC), and/or a thermal energy load disturbance (i.e., a heat load disturbance) {dot over (Q)}_(other).

In some embodiments, the model predictive controller generates an HVAC load model. The HVAC load model may describe the amount of heating or cooling {dot over (Q)}_(HVAC) provided by the HVAC equipment as a function of the temperature setpoint T_(sp) and the temperature of the air T_(ia) within the building zone. The model predictive controller can use the thermal mass storage model and the HVAC load model to predict system states (i.e., building mass temperature T_(m), building air temperature T_(ia), etc.), predict the load disturbance {dot over (Q)}_(other), and establish constraints on the optimization performed by the model predictive controller.

In some embodiments, the model predictive controller uses the thermal mass storage model and the HVAC load model to take advantage of the building mass as a thermal energy storage medium. For example, thermal energy can be stored in the solid building mass (i.e., precooling or preheating the building zone) to allow the model predictive controller to take advantage of time-varying utility rates and demand charges when determining the optimal temperature setpoints T_(sp). To precool the building zone, thermal energy may be removed from the thermal mass during periods when energy prices are lowest. During periods when energy prices are largest, thermal energy may be moved back into the solid mass from the zone air, which reduces the sensible HVAC load {dot over (Q)}_(HVAC) needed to maintain the zone air within a comfortable temperature range. Similarly, the model predictive controller can preheat the solid mass within the building zone when energy prices are low and use the stored thermal energy to warm the zone air when energy prices are higher.

In some embodiments, the model predictive controller is configured to operate in two distinct modes: (1) parameter identification mode and (2) operational mode. In the parameter identification mode, the model predictive controller can manipulate the temperature setpoint T_(sp) provided to the smart thermostat and/or the equipment controller to induce a dynamic response in the zone temperature T_(ia). This procedure of inducing a dynamic response is referred to as the parameter identification experiment or system identification process and can be used to identify the values of the parameters in the thermal mass storage model and the HVAC load model. In the operational mode, the model predictive controller can solve an optimal control problem to determine the optimal temperature setpoint trajectory T_(sp) (i.e., a time series of temperature setpoints) that minimizes the cost of energy consumed by the HVAC equipment over the duration of the optimization period. The optimal temperature setpoints T_(sp) can be provided to the smart thermostat and/or the equipment controller for use in generating operating commands for the HVAC equipment.

The setpoint T_(sp) computed for the first time step of the optimization period can be sent to the smart thermostat and/or the equipment controller to be implemented during the first time step. The smart thermostat and/or the equipment controller can turn stages of HVAC equipment on and/or off such that the temperature T_(ia) of the zone is forced to and then maintained at the temperature setpoint T_(sp). At the next time step, the model predictive controller receives updated information (i.e., feedback on the zone conditions), resolves the optimal control problem, and sends the temperature setpoint T_(sp) for the next time step to the smart thermostat and/or the equipment controller.

The model predictive controller can collect feedback on the zone conditions (e.g., zone temperature T_(ia), outdoor air temperature T_(oa), heating/cooling load {dot over (Q)}_(HVAC)) when operating in the parameter identification mode. The zone conditions can be monitored and recorded over the identification experiment, along with the commanded zone temperature setpoint T_(sp), as input-output data (i.e., training data). The model predictive controller can use the input-output data to compute the zone thermal model parameters by performing a system identification process. The feedback received from the smart thermostat may include the current minimum and maximum allowable setpoint to limit the adjustments to the zone temperature setpoint T_(sp) made during the identification experiment. These and other features of the smart thermostat and/or the model predictive controller are described in greater detail below.

Throughout this disclosure, the terms “optimal,” “optimized,” “optimum,” and the like are used to refer to values (e.g., temperature setpoint values, HVAC load values, etc.) that are determined by performing an optimization process. Similarly, the term “optimizing” is used to refer to the process of performing an optimization. In some instances, the values resulting from the optimization process are true optimal values (i.e., values that achieve the minimum possible value or maximum possible value for a performance variable given the constraints on the optimization process). In other instances, the values resulting from the optimization process are not true optimal values. This can result from imperfect information used to perform the optimization, inaccuracies in the predictive model used to constrain the optimization, and/or various other factors that can prevent the optimization from converging on the true optimal values. The terms “optimal,” “optimized,” “optimum,” and the like should be interpreted to include any values that result from performing an optimization, regardless of whether the values are true optimal values. Similarly, the term “optimizing” should be interpreted to include the process of performing an optimization, regardless of whether that optimization converges on the true optimal values.

Smart Thermostat

Referring now to FIGS. 1-2, a smart thermostat 100 is shown, according to some embodiments. Smart thermostat 100 can be configured to monitor and control one or more environmental conditions of a building zone (e.g., temperature, humidity, air quality, etc.). In some embodiments, smart thermostat 100 is mounted on a wall within the building zone and configured to measure the temperature, humidity, and/or other environmental conditions of the building zone. Smart thermostat 100 can be configured to communicate with HVAC equipment that operates to affect the measured environmental conditions.

Smart thermostat 100 can be installed in a home, office building, school, hospital, or any other environment-controlled space. In residential implementations, the HVAC equipment controlled by smart thermostat 100 may include a home furnace, air conditioning unit, and/or other types of residential HVAC equipment. In commercial implementations, the HVAC equipment may include one or more chillers, boilers, air handling units, rooftop units, dampers, or other types of HVAC equipment configured to affect the environment of a building zone. It is contemplated that smart thermostat 100 can be configured to control any type of HVAC equipment.

In some embodiments, smart thermostat 100 includes a communications interface that enables smart thermostat 100 to connect to a communications network (e.g., a local area network, the Internet, a cellular network, etc.) and communicate with various external systems and devices (e.g., user devices, remote servers, remote controllers, etc.). For example, smart thermostat 100 can be configured to receive weather forecasts from a remote weather service via the Internet. In some embodiments, smart thermostat 100 receives temperature setpoints from a model predictive controller via the communications network. In other embodiments, the model predictive controller is a component of smart thermostat 100.

Smart thermostat 100 is shown to include a display screen 102 and a base 104. Base 104 can attach to a wall or other surface upon which smart thermostat 100 is mounted. Display screen 102 may be transparent or semi-transparent (e.g., an organic LED display) and can be configured to display text, graphics, and other information for presentation to a user. In some embodiments, display screen 102 is touch-sensitive (e.g., a capacitive or resistive touch screen) and configured to receive user input. Display screen 102 may have a landscape aspect ratio as shown in FIG. 1 (i.e., the width of display screen 102 may exceed the height of display screen 102) or a portrait aspect ratio as shown in FIG. 2 (i.e., the height of display screen 102 may exceed the width of display screen 102). Display screen 102 may be cantilevered from base 104 (i.e., attached to base 104 along a single edge of display screen 102) and may extend from base 104 in a direction substantially parallel to the mounting surface.

Smart thermostat 100 can include some of all of the features of the thermostats described in U.S. patent application Ser. No. 15/143,373 filed Apr. 29, 2016, U.S. patent application Ser. No. 15/146,763 filed May 4, 2016, U.S. patent application Ser. No. 15/146,749 filed May 4, 2016, U.S. patent application Ser. No. 15/146,202 filed May 4, 2016, U.S. patent application Ser. No. 15/146,134 filed May 4, 2016, U.S. patent application Ser. No. 15/146,649 filed May 4, 2016, U.S. Provisional Patent Application No. 62/331,863 filed May 4, 2016, U.S. Provisional Patent Application No. 62/352,955 filed Jun. 21, 2016, U.S. patent application Ser. No. 15/298,191 filed Oct. 19, 2016, U.S. patent application Ser. No. 15/336,793 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/336,792 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/336,789 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/338,221 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/338,215 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/336,791 filed Oct. 28, 2016, U.S. patent application Ser. No. 15/397,722 filed Jan. 3, 2017, and/or U.S. Provisional Patent Application No. 62/446,296 filed Jan. 13, 2017. The entire disclosure of each of these patent applications is incorporated by reference herein.

Model Predictive Control Systems

Referring now to FIGS. 3-4, a pair of model predictive control (MPC) systems 300 and 400 are shown, according to some embodiments. MPC system 300 is shown to include a model predictive controller 302, a communications network 304, smart thermostat 100, HVAC equipment 308, and a building zone 310. In MPC system 300, model predictive controller 302 is separate from smart thermostat 100 and configured to provide a temperature setpoint T_(sp) to smart thermostat 100 via communications network 304. In MPC system 400, model predictive controller 302 is a component of smart thermostat 100 along with an equipment controller 406.

Building zone 310 can include one or more rooms or zones within a home, office building, school, hospital, or any other environment-controlled space. HVAC equipment 308 can include any type of equipment operable to affect the temperature, humidity, and/or other environmental conditions of building zone 310. For example, HVAC equipment 308 can include a home furnace, air conditioning unit, one or more chillers, boilers, air handling units, rooftop units, dampers, or other types of HVAC equipment configured to affect the environment of building zone 310. In some embodiments, other building devices (e.g., lighting, window shades) can be controlled by the smart thermostat 100 to affect heat disturbances at the building zone 310.

Model predictive controller 302 can be configured to determine an optimal temperature setpoint T_(sp) for smart thermostat 100 for each of a plurality of time steps during an optimization period. Smart thermostat 100 can use the temperature setpoints T_(sp) provided by model predictive controller 302 to generate equipment commands for HVAC equipment 308. The equipment commands can be generated by smart thermostat 100 and/or equipment controller 406 (e.g., an on/off controller, an equipment staging controller, a proportional-integral (PI) controller, a proportional-integral-derivative (PID) controller, etc.). HVAC equipment 308 operate according to the equipment commands to provide variable amount of heating or cooling {dot over (Q)}_(HVAC) to building zone 310. By controlling the temperature setpoint T_(sp), model predictive controller 302 can modulate the amount of heating or cooling {dot over (Q)}_(HVAC) provided by HVAC equipment 308, thereby affecting the temperature of building zone 310.

To determine the optimal temperature setpoints T_(sp), model predictive controller 302 can optimize an objective function (i.e., a cost function) that accounts for the cost of operating HVAC equipment 308 over the duration of the optimization period. The costs of operating HVAC equipment 308 can include, for example, the costs of resources consumed by HVAC equipment 308 during operation (e.g., electricity, natural gas, water, etc.), demand charges imposed by an electric utility, peak load contribution charges, equipment degradation/replacement costs, and/or other costs associated with the operation of HVAC equipment 308. The optimization performed by model predictive controller 302 is described in greater detail with reference to FIG. 5.

Model predictive controller 302 can optimize the objective function subject to a set of constraints. The constraints may include temperature constraints (e.g., a maximum temperature limit and a minimum temperature limit for building zone 310), equipment capacity constraints, load change constraints, thermal mass storage constraints, HVAC load constraints, and/or other constraints that limit the operation of HVAC equipment 308 and/or describe the temperature evolution of building zone 310. Model predictive controller 302 can automatically generate the optimization constraints by performing a system identification process and generating predictive models that describe the controlled system (i.e., HVAC equipment 308 and building zone 310).

In some embodiments, model predictive controller 302 generates a thermal mass storage model that describes the temperature of the air within building zone 310 T_(ia) as a function of several physical parameters and variables. The physical parameters in the thermal mass storage model may include, for example, the thermal capacitance C_(ia) of the air in building zone 310, the thermal capacitance C_(m) of the solid mass within building zone 310, the thermal resistance R_(mi) between the air within building zone 310 and the solid mass, the thermal resistance R_(oi) between the air within building zone 310 and the ambient environment, etc. The variables in the thermal mass storage model may include the temperature of air within building zone 310 T_(ia), the temperature of the solid mass within building zone 310 T_(m), the outside air temperature T_(oa), the amount of heating or cooling provided by HVAC equipment 308 {dot over (Q)}_(HVAC), and/or a thermal energy load disturbance {dot over (Q)}_(other).

In some embodiments, model predictive controller 302 generates a HVAC load model. The HVAC load model may describe the amount of heating or cooling {dot over (Q)}_(HVAC) provided by HVAC equipment 308 as a function of the temperature setpoint T_(sp) and the temperature of the air T_(ia) within building zone 310. Model predictive controller 302 can use the thermal mass storage model and the HVAC load model to predict system states (i.e., building mass temperature T_(m), building air temperature T_(ia), etc.), predict the load disturbance {dot over (Q)}_(other), and establish constraints on the optimization performed by model predictive controller 302.

In some embodiments, model predictive controller 302 uses the thermal mass storage model and the HVAC load model to take advantage of the building mass as a thermal energy storage medium. For example, thermal energy can be stored in the solid building mass (i.e., precooling or preheating building zone 310) to allow model predictive controller 302 to take advantage of time-varying utility rates and demand charges when determining the optimal temperature setpoints T_(sp). To precool building zone 310, thermal energy may be removed from the thermal mass during periods when energy prices are lowest. During periods when energy prices are largest, thermal energy may be moved back into the solid mass from the zone air, which reduces the sensible HVAC load {dot over (Q)}_(HVAC) needed to maintain the zone air within a comfortable temperature range. Similarly, model predictive controller 302 can preheat the solid mass within building zone 310 when energy prices are low and use the stored thermal energy to warm the zone air when energy prices are higher. In some embodiments, the model predictive controller 302 is configured to provide setpoints for window shades and/or lighting devices, allowing the model predictive controller 302 to affect an amount of heat provided to the building zone by solar irradiance and/or by the operation of lighting devices in the building zone.

In some embodiments, model predictive controller 302 is configured to operate in two distinct modes: (1) parameter identification mode and (2) operational mode. In the parameter identification mode, model predictive controller 302 can manipulate the temperature setpoint T_(sp) provided to smart thermostat 100 and/or equipment controller 406 to induce a dynamic response in the zone temperature T_(ia). This procedure of inducing a dynamic response is referred to as the parameter identification experiment or system identification process and can be used to identify the values of the parameters in the thermal mass storage model and the HVAC load model. In the operational mode, model predictive controller 302 can solve an optimal control problem to determine the optimal temperature setpoint trajectory T_(sp) (i.e., a time series of temperature setpoints) that minimizes the cost of energy consumed by HVAC equipment 308 over the duration of the optimization period. The optimal temperature setpoints T_(sp) can be provided to smart thermostat 100 and/or equipment controller 406 for use in generating operating commands for HVAC equipment 308.

The setpoint T_(sp) computed for the first time step of the optimization period can be sent to smart thermostat 100 and/or equipment controller 406 to be implemented during the first time step. Smart thermostat 100 and/or equipment controller 406 can turn stages of HVAC equipment 308 on and/or off such that the temperature T_(ia) of the zone is forced to and then maintained at the temperature setpoint T_(sp). At the next time step, model predictive controller 302 receives updated information (i.e., feedback on the zone conditions), resolves the optimal control problem, and sends the temperature setpoint T_(sp) for the next time step to smart thermostat 100 and/or equipment controller 406.

Model predictive controller 302 can collect feedback on the zone conditions (e.g., zone temperature T_(ia), outdoor air temperature T_(oa), heating/cooling load {dot over (Q)}_(HVAC)) when operating in the parameter identification mode. The zone conditions can be monitored and recorded over the identification experiment, along with the commanded zone temperature setpoint T_(sp), as input-output data (i.e., training data). Model predictive controller 302 can use the input-output data to compute the zone thermal model parameters by performing a system identification process. The feedback received from smart thermostat 100 may include the current minimum and maximum allowable setpoint to limit the adjustments to the zone temperature setpoint T_(sp) made during the identification experiment.

In some embodiments, a pseudo-random binary signal (PRBS) is generated that is subsequently used to produce a signal that takes values in the set {T_(sp,1), T_(sp,2)}, where T_(sp,1) and T_(sp,2) are maximum and minimum allowable temperature setpoints depending on mode of operation of smart thermostat 100 (e.g., heating, cooling, home, sleep, or away mode). The minimum and maximum temperature setpoints may be time-varying to account for the different operational modes of smart thermostat 100 (e.g., home, sleep, and away). In some embodiments, smart thermostat 100 uses occupancy detection and/or user feedback to change the mode.

To determine the optimal temperature setpoints T_(sp) in the operational mode, model predictive controller 302 can use the thermal model of building zone 310 to predict the zone temperature T_(ia) and HVAC fuel consumption over the optimization period, given a forecast of the weather and heat disturbance load on the zone (e.g., heat generated from people and electrical equipment and gained through solar radiation). The zone temperature T_(ia) may be subject to comfort constraints. The predicted trajectories of the zone temperature T_(ia), temperature setpoint T_(sp), and HVAC fuel consumption can be sent to smart thermostat 100 to be displayed to the user allowing the user to better understand the setpoint decision made by model predictive controller 302.

In some embodiments, smart thermostat 100 allows a user to override the optimal temperature setpoints computed by model predictive controller 302. For example, the smart thermostat 100 may operate the equipment 308 using the user-input temperature setpoint for a preset amount of time after the user input is received before switching back to the optimal temperature setpoints. As another example, a user-input temperature setpoint or schedule remains in control until an instruction is received from the user to use the optimal temperature setpoints from the model predictive controller 302. As another example, the user-input temperature setpoint may be used to define one or more constraints on an optimization performed by the model predictive controller 302. For example, a predicted temperature may be constrained to be within a preset error of the temperature setpoint input by the user while the model predictive controller 302 continues to operate as described in greater detail below.

Smart thermostat 100 may include a savings estimator configured to determine a potential cost savings resulting from the optimal temperature setpoints relative to the user-specified temperature setpoints. Smart thermostat 100 can be configured to display the potential cost savings via display screen 102 to inform the user of the economic cost predicted to result from overriding the optimal temperature setpoints. These and other features of model predictive controller 302 are described in greater detail below.

Model Predictive Controller

Referring now to FIG. 5, a block diagram illustrating model predictive controller 302 in greater detail is shown, according to come embodiments. Model predictive controller 302 is shown to include a communications interface 502 and a processing circuit 504. Communications interface 502 may facilitate communications between model predictive controller 302 and external systems or devices. For example, communications interface 502 may receive measurements of zone conditions from smart thermostat 100, HVAC equipment 308, and/or building zone 310. Zone conditions may include the temperature T_(ia) of the air within building zone 310, the outside air temperature T_(oa), and/or the amount of heating/cooling {dot over (Q)}_(HVAC) provided to building zone 310 at each time step of the optimization period. Communications interface 502 may also receive temperature constraints for building zone 310 and/or equipment constraints for HVAC equipment 308. In some embodiments, communications interface 502 receives utility rates from utilities 524 (e.g., time-varying energy prices, demand charges, etc.), weather forecasts from a weather service 526, and/or historical load and rate data from historical data 528. In some embodiments, model predictive controller 302 uses communications interface 502 to provide temperature setpoints T_(sp) to smart thermostat 100 and/or equipment controller 406.

Communications interface 502 may include wired or wireless communications interfaces (e.g., jacks, antennas, transmitters, receivers, transceivers, wire terminals, etc.) for conducting data communications external systems or devices. In various embodiments, the communications may be direct (e.g., local wired or wireless communications) or via a communications network (e.g., a WAN, the Internet, a cellular network, etc.). For example, communications interface 502 can include an Ethernet card and port for sending and receiving data via an Ethernet-based communications link or network. In another example, communications interface 502 can include a Wi-Fi transceiver for communicating via a wireless communications network or cellular or mobile phone communications transceivers.

Processing circuit 504 is shown to include a processor 506 and memory 508. Processor 506 may be a general purpose or specific purpose processor, an application specific integrated circuit (ASIC), one or more field programmable gate arrays (FPGAs), a group of processing components, or other suitable processing components. Processor 506 is configured to execute computer code or instructions stored in memory 508 or received from other computer readable media (e.g., CDROM, network storage, a remote server, etc.).

Memory 508 may include one or more devices (e.g., memory units, memory devices, storage devices, etc.) for storing data and/or computer code for completing and/or facilitating the various processes described in the present disclosure. Memory 508 may include random access memory (RAM), read-only memory (ROM), hard drive storage, temporary storage, non-volatile memory, flash memory, optical memory, or any other suitable memory for storing software objects and/or computer instructions. Memory 508 may include database components, object code components, script components, or any other type of information structure for supporting the various activities and information structures described in the present disclosure. Memory 508 may be communicably connected to processor 506 via processing circuit 504 and may include computer code for executing (e.g., by processor 506) one or more processes described herein. When processor 506 executes instructions stored in memory 508 for completing the various activities described herein, processor 506 generally configures controller 302 (and more particularly processing circuit 504) to complete such activities.

Still referring to FIG. 5, model predictive controller 302 is shown to include a system identifier 510, a load/rate predictor 518, a state/disturbance estimator 520, and a predictive optimizer 522. In brief overview, system identifier 510 can be configured to perform a system identification process to identify the values of the parameters in the building thermal mass model and the HVAC load model. The parameters identified by system identifier 510 may define the values of the system matrices A, B, C, and D and the estimator gain K. State/disturbance estimator 520 can be configured to compute an estimate of the current state {circumflex over (x)}(k) and unmeasured disturbance {circumflex over (d)}(k). The estimation performed by state/disturbance estimator 520 can be performed at each sample time and may use the system matrices A, B, C, D and estimator gain K identified by system identifier 510.

Load/rate predictor 518 can generate predictions or forecasts of the internal heat load {dot over (Q)}_(other), the non-HVAC building electricity consumption (used to calculate electrical demand charges), and the utility rates at each time step of the optimization period as a function of historical data 528 and several key predictor variables (e.g., time-of-day, outdoor air conditions, etc.). The models used by load/rate predictor 518 may be different than that identified by system identifier 510. In some embodiments, load/rate predictor 518 uses the prediction techniques described in U.S. patent application Ser. No. 14/717,593 filed May 20, 2015, the entire disclosure of which is incorporated by reference herein.

Predictive optimizer 522 can manipulate the temperature setpoint T_(sp) of building zone 310 to minimize the economic cost of operating HVAC equipment 308 over the duration of the optimization period using the building thermal mass storage model, the HVAC load model, and state and disturbance estimates and forecasts. In some embodiments, predictive optimizer 522 uses open-loop predictions to optimize the temperature setpoint T_(sp) trajectory over the optimization period. At each time step of the optimization period, the predictions can be updated based on feedback from the controlled system (e.g., measured temperatures, measured HVAC loads, etc.).

Before discussing system identifier 510, load/rate predictor 518, state/disturbance estimator 520, and predictive optimizer 522 in detail, the notation used throughout the remainder of the present disclosure and the class of system model used by model predictive controller 302 are explained. The set of integers is denoted by

and the set of positive integers is denoted by

_(≥0), and the set of integers contained in the interval [a, b] is denoted by

_(a:b). For a time-dependent vector x(k)∈

^(n), {circumflex over (x)}(i|k)∈

^(n) denotes the estimated value of x at time step i∈

_(≥0) given the measurement at time step k∈

_(≥0) where i≥k and {tilde over (x)}(i|k)∈

^(n) denotes the (open-loop) predicted value of x at time step i∈

_(≥0) with the prediction initialized at time k∈

_(≥0) (i≥k). For notational simplicity, the notation of the predicted value of x at time i starting from time k is abbreviated to {tilde over (x)}(i). Boldface letters are used to represent a sequence with cardinality N∈

_(≥0) (i.e., x:={x(0) . . . x(N−1)}). The notation ⋅* (e.g., x*) denotes an optimal quantity with respect to some optimization problem.

In some embodiments, the model generated and used by model predictive controller 302 is a discrete-time linear time-invariant model, as shown in the following equation: x(k+1)=Ax(k)+Bu(k) y(k)=Cx(k)+Du(k)  (Equation 1) where k∈

_(≥0) is the time index, x(k)∈

^(n) is the state vector, y(k)∈

^(p) is the measured output, and u(k)∈

^(m) is the input vector. The matrix A can be assumed to be stable; that is the real parts of the eigenvalues of A lie within the unit circle. The pair (A, B) can be assumed to be controllable and the pair (A, C) can be assumed to be observable. Some HVAC equipment 308 and devices are only operated at finite operating stages. In the building models considered, this gives rise to discrete outputs whereby the measured output takes values within a finite set of numbers: y_(i)(k)∈{y_(i,stage,1), y_(i,stage,2), . . . , y_(i,stage,m)} for some i. In this case, the output does not evolve according to Equation 1.

In some embodiments, the model generated and used by model predictive controller 302 is a grey box continuous-time models derived from thermal resistance-capacitance (RC) modeling principals. Model predictive controller 302 can be configured to parameterize the resulting continuous-time thermal models and identify the continuous-time model parameters. The continuous-time version of the model of Equation 1 can be represented as shown in the following equation {dot over (x)}(t)=A _(c) x(t)+B _(c) u(t) y(t)=C _(c) x(t)+D _(c) u(t)  (Equation 2) where t≥0 is continuous time (the initial time is taken to be zero) and A_(c), B_(c), C_(c), D_(c) are the continuous-time versions of the discrete-time system matrices A, B, C, and D. The values of the parameters in the matrices A, B, C, and D as well as an estimator gain K can be determined by system identifier 510 by performing a system identification process (described in greater detail below) State/Disturbance Estimator

State/disturbance estimator 520 can be configured to compute an estimate of the current state {circumflex over (x)}(k) and unmeasured disturbance {circumflex over (d)}(k). The estimation performed by state/disturbance estimator 520 can be performed at each sample time (i.e., when operating in the operational mode) and may use the system matrices A, B, C, D and estimator gain K identified by system identifier 510. In some embodiments, state/disturbance estimator 520 modifies the system model to account for measurement noise and process noise, as shown in the following equation: x(k+1)=Ax(k)+Bu(k)+w(k) y(k)=Cx(k)+Du(k)+v(k)  (Equation 3) where w(k) is the process noise and v(k) is the measurement noise.

State/disturbance estimator 520 can use the modified system model of Equation 3 to estimate the system state {circumflex over (x)}(k) as shown in the following equation: {circumflex over (x)}(k+1|k)=A{circumflex over (x)}(k|k−1)+Bu(k)+K(y(k)−ŷ(k|k−1)) ŷ(k|k−1)=C{circumflex over (x)}(k|k−1)+Du(k)  (Equation 4) where {circumflex over (x)}(k+1|k) is the estimated/predicted state at time step k+1 given the measurement at time step k, ŷ(k|k−1) is the predicted output at time step k given the measurement at time step k−1, and K is the estimator gain. Equation 4 describes the prediction step of a Kalman filter. The Kalman filter may be the optimal linear filter under some ideal assumptions (e.g., the system is linear, no plant-model mismatch, the process and measurement noise are follow a white noise distribution with known covariance matrices).

In some embodiments, the estimated state vector {circumflex over (x)}(k+1|k), the output vector ŷ(k|k−1), and the input vector u(k) are defined as follows:

${{\hat{x}\left( {k + 1} \middle| k \right)} = \begin{bmatrix} {{\hat{T}}_{ia}\left( {k + 1} \middle| k \right)} \\ {{\hat{T}}_{m}\left( {k + 1} \middle| k \right)} \\ {\hat{I}\left( {k + 1} \middle| k \right)} \end{bmatrix}},{{\hat{y}\left( k \middle| {k - 1} \right)} = \begin{bmatrix} {{\hat{T}}_{ia}\left( k \middle| {k - 1} \right)} \\ {{\overset{.}{\hat{Q}}}_{HVAC}\left( k \middle| {k - 1} \right)} \end{bmatrix}},{{u(k)} = \begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}$ where {circumflex over (T)}_(ia) is an estimate of the zone air temperature T_(ia), {circumflex over (T)}_(m) is an estimate of the zone mass temperature T_(m), Î is an estimate of the integrating disturbance I, {circumflex over ({dot over (Q)})}_(HVAC) is an estimate of the heating or cooling load provided by HVAC equipment 308, T_(sp) is the temperature setpoint, and T_(oa) is the outdoor air temperature.

The state estimation described by Equation 4 is effective in addressing stationary noise. However, unmeasured correlated disturbances are also common in practice. Correlated disturbances can lead to biased estimates/predictions. In some embodiments, state/disturbance estimator 520 accounts for disturbances by adding a disturbance state to the system model, as shown in the following equation:

$\begin{matrix} {\begin{bmatrix} {\hat{x}\left( {k + 1} \middle| k \right)} \\ {\hat{d}\left( {k + 1} \middle| k \right)} \end{bmatrix} = {\begin{bmatrix} A & B_{d} \\ 0 & I \end{bmatrix}{\quad{{\begin{bmatrix} {\hat{x}\left( {k + 1} \middle| k \right)} \\ {\hat{d}\left( {k + 1} \middle| k \right)} \end{bmatrix} + {\begin{bmatrix} B \\ 0 \end{bmatrix}{u(k)}} + {\begin{bmatrix} K_{x} \\ K_{d} \end{bmatrix}\left( {{y(k)} - {\hat{y}\left( k \middle| {k - 1} \right)}} \right)\mspace{79mu}{\hat{y}\left( k \middle| {k - 1} \right)}}} = {\begin{bmatrix} C & C_{d} \end{bmatrix}{\quad{\begin{bmatrix} {\hat{x}\left( k \middle| {k - 1} \right)} \\ {\hat{d}\left( k \middle| {k - 1} \right)} \end{bmatrix} + {{Du}(k)}}}}}}}} & \left( {{Equation}\mspace{14mu} 5} \right) \end{matrix}$ where {circumflex over (d)}(k) is the unmeasured disturbance at time step k. The effect of adding the disturbance state is similar to that of the integral action of a proportional-integral controller. State/disturbance estimator 520 may select values of B_(d) and C_(a) and modify the estimation problem as shown in Equation 5. In some embodiments, state/disturbance estimator 520 selects values of B_(d) and C_(d) such that the system model is an integrating input and/or output disturbance model.

In some embodiments, the choice of disturbance models may have a significant impact on the ability of model predictive controller 302 to adequately estimate the lumped thermal building mass temperature T_(m). Accurate estimation of the thermal mass temperature T_(m) enables model predictive controller 302 to effectively use the solid building mass to store thermal energy when preheating or precooling building zone 310 in order to time-shift the thermal energy load and minimize the cost of operating HVAC equipment 308.

System Identifier

System identifier 510 can be configured to perform a system identification process to identify the values of the parameters in the building thermal mass model and the HVAC load model. The functions performed by system identifier 510 may be performed when operating in the parameter identification mode. The parameters identified by system identifier 510 may define the values of the system matrices A, B, C, and D and the estimator gain K. System identifier 510 can perform the system identification process during the commissioning phase of the control system and/or after the detection of a substantial change in the model parameters that has resulted in significant closed-loop performance deterioration. In some embodiments, the system identification process performed by system identifier 510 is the same or similar to the system identification process described in U.S. Pat. No. 9,235,657 titled “System Identification and Model Development” and granted Jan. 12, 2016. The entire disclosure of U.S. Pat. No. 9,235,657 is incorporated by reference herein.

System identification (SI) is the art and science of building mathematical models of dynamic systems from observed input-output data. In general, a model for a system may be developed using three paradigms: white box, grey box or black box models. These three system identification modeling paradigms are summarized in FIG. 6.

For applications that require high fidelity models, white box models may be derived via detailed first-principles modeling approaches. This approach often requires a high degree of knowledge of the physical system and may result in a high-order nonlinear dynamic model. While the resulting model may capture most of the relevant physical behavior a system, there is usually a very high engineering investment required to develop such a model. Given the complexities of the resulting models, often white box models cannot be used in real-time predictive controllers owing to real-time computational restrictions. Nevertheless, white box models may help to evaluate control approaches in simulation to identify and mitigate potential pitfalls of the approach.

In a black box modeling paradigm, little a priori knowledge about the model structure is assumed except for standard assumptions like linearity and stability of the model. Black box methods fit the best model to the input-output data. While there is little to no assumed a priori knowledge about the model, which offers a deal of flexibility, there are a few drawbacks to the approach. Given that little about the model structure is assumed, black box models often tend to have a large amount of parameters, which requires a large amount of data to properly select the model order and appropriate input-output representation. To capture the necessary amount of input-output data needed for black box methods, an undesirable length for the system identification experiment may be required.

Finally, grey box modeling assumes that there is a known model structure often derived through simplified first-principals or semi-physical modeling. The model structure is parameterized and parameter estimation techniques are used to estimate the parameters. Owing to the incorporation of more a priori knowledge into grey box models, the resulting number of parameters for these types of models are often less than black box models. As a result of fewer parameters as well as the fact that the model structure (given it is valid) already encodes important relationships, less input-output data is typically needed to fit a good model with respect to that needed to fit a good model for black box methods.

In some embodiments, system identifier 510 uses a grey box model to represent HVAC equipment 308 and building zone 310 (i.e., the controlled system). In a grey box model system identification approach, a model structure

is selected, which is subsequently parameterized with a parameter θ∈D

⊂

^(d) where D

is the set of admissible model parameter values. The resulting possible models are given by the set: M={

(θ)|θ∈D

}  (Equation 6)

To fit a proper parameter vector to the controlled system, system identifier 510 can collect input-output data. The input-output data may include samples of the input vector u(k) and corresponding samples of the output vector y(k), as shown in the following equation: Z ^(N)=[y(1),u(1),y(2),u(2), . . . ,y(N),u(N)]  (Equation 7) where N>0 is the number of samples collected. Parameter estimation is the problem of selecting the proper value {circumflex over (θ)}_(N) of the parameter vector given the input-output data Z^(N) (i.e., it is the mapping: Z^(N)→{circumflex over (θ)}_(N) ∈D

).

In some embodiments, system identifier 510 uses a prediction error method to select the values of the parameter vector {circumflex over (θ)}_(N). A prediction error method (PEM) refers to a particular family of parameter estimation methods that is of interest in the context of the present disclosure. The prediction error method used by system identifier 510 may include fitting the parameter vector {circumflex over (θ)}_(N) by minimizing some function of the difference between the predicted output and observed output. For example, let ŷ(k, θ) be the predicted output at time step k given the past input-output sequence Z^(k-1) and the model

(θ) for θ∈D

. The prediction error at time step k is given by: ε(k,θ):=y(k)−ŷ(k,θ)  (Equation 8)

In some embodiments, system identifier 510 filters the prediction error sequence through a stable linear filter and defines the following prediction performance metric:

$\begin{matrix} {{{V_{N}\left( {\theta,Z^{N}} \right)} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}{\ell\left( {ɛ_{F}\left( {k,\theta} \right)} \right)}}}}\mspace{11mu}} & \left( {{Equation}\mspace{14mu} 9} \right) \end{matrix}$ where l(⋅) is the cost function (e.g., a positive definite function) and ε_(F)(k, θ) is the filtered prediction error. The parameter estimate {circumflex over (θ)}_(N) of a prediction error method is then given by: {circumflex over (θ)}_(N)={circumflex over (θ)}_(N)(Z ^(N))=

V _(N)(θ,Z ^(N))  (Equation 10)

In various embodiments, system identifier 510 can use any parameter estimation method that corresponds to Equation 10. For example, system identifier 510 can define a quadratic cost function that is the square of the prediction errors, as shown in the following equation: l(ε(k,θ))=l(y(k)−ŷ(k|k−1,θ))=∥y(k)−ŷ(k|k−1,θ)∥₂ ²  (Equation 11) where ŷ(k|k−1, θ) denotes the one-step ahead prediction of the output using the model

(θ). When the prediction errors are independently and identically distributed random variables from a normal distribution (i.e., the process and measurement noise is Gaussian white noise) and the model being identified is linear, the cost function of Equation 11 is optimal from a statistical point-of-view.

Still referring to FIG. 5, system identifier 510 is shown to include a thermal mass storage model generator 512, a HVAC load model generator 514, and an estimator gain identifier 516. Thermal mass storage model generator 512 can be configured to estimate or identify the parameters of a thermal mass storage model that describes the relationship between indoor air temperature T_(ia), building mass temperature T_(m), outdoor air temperature T_(oa), internal load {dot over (Q)}_(other), and sensible HVAC load {dot over (Q)}_(HVAC). The indoor air temperature T_(ia) represents the temperature of the air within building zone 310. The building mass temperature T_(m) represents the temperature of the solid objects within building zone 310. The internal load {dot over (Q)}_(other) represents heat generated internally within building zone 310 by building occupants, electronics, resistive loads, and other sources of heat. The sensible HVAC load {dot over (Q)}_(HVAC) represents the amount of heating or cooling applied to building zone 310 by HVAC equipment 308.

HVAC load model generator 514 can be configured to generate a HVAC load model that describes the relationship between indoor air temperature T_(ia), the temperature setpoint T_(sp), and the sensible HVAC load {dot over (Q)}_(HVAC). In some embodiments, the HVAC load model describes the dynamics of HVAC equipment 308 when providing heating or cooling to building zone 310. For example, the HVAC load model may represent HVAC equipment 308 as a proportional-integral (PI) control system that provides the sensible HVAC load {dot over (Q)}_(HVAC) as a function of the indoor air temperature T_(ia) and the temperature setpoint T_(sp).

Estimator gain identifier 516 can be configured to calculate the state/disturbance estimation gain (i.e., the estimator gain K). The estimator gain K may be referred to in other contexts as the disturbance matrix or Kalman filter gain. The operation of thermal mass storage model generator 512, HVAC load model generator 514, and estimator gain identifier 516 are described in greater detail below.

Thermal Mass Storage Model Generator

Thermal mass storage model generator 512 can be configured to estimate or identify the parameters of a thermal mass storage model that describes the relationship between indoor air temperature T_(ia), building mass temperature T_(m), outdoor air temperature T_(oa), internal load {dot over (Q)}_(other), and sensible HVAC load {dot over (Q)}_(HVAC). In some embodiments, thermal mass storage model generator 512 performs a multi-stage system identification process to generate the parameters of the thermal mass storage model. The stages of the system identification process may include input-output data collection, generating a parameterized building thermal model, and estimating (i.e., fitting) the parameters of the parameterized building thermal model using a parameter identification algorithm.

The first stage of the system identification process is input-output data collection. The type of input-output data collected to fit the model parameters may have a significant impact in the usefulness and accuracy of the resulting model. In some embodiments, thermal mass storage model generator 512 manipulates the temperature setpoint T_(sp) in an open-loop fashion as to sufficiently and persistently excite the system. For example, thermal mass storage model generator 512 can persistently excite (PE) a signal {s(k):k=0, 1, 2, . . . } with spectrum ϕ_(s)(ω) if ϕ_(s)(ω)>0 for almost all ω where:

$\begin{matrix} {{{\phi_{s}(\omega)} = {\sum\limits_{m = {- \infty}}^{\infty}\left( {\lim\limits_{N\rightarrow\infty}{\frac{1}{N}{\sum\limits_{k = 1}^{N}{{s(k)}{s\left( {k - m} \right)}}}}} \right)}}\mspace{11mu}} & \left( {{Equation}\mspace{14mu} 12} \right) \end{matrix}$

Input-output data collected during closed-loop operation may not provide an information-rich dataset with respect to system identification because the inputs tend to be correlated to themselves as well as to the unmeasured disturbances as a result of closing the loop with a controller. As a result input-output data obtained via closed-loop operation may not meet the PE condition without adding perturbations to the inputs similar to that used in extremum seeking control.

Instead of closed-loop identification, thermal mass storage model generator 512 can perform a system identification experiment is to meet the PE condition. In some embodiments, thermal mass storage model generator 512 performs the system identification experiment while building zone 310 is occupied. Accordingly, it may be desirable to design the system identification experiment to be as short as possible in order to prevent or minimize disruptions with building occupants. In some embodiments, thermal mass storage model generator 512 makes small changes to the temperature setpoints T_(sp) (e.g., changes of 3-5° F.) in order to have minimal impact on the comfort of the building occupants.

In some embodiments, thermal mass storage model generator 512 generates the excitation signals using a pseudorandom binary sequence (PRBS). FIG. 7 shows a graph 700 of an example excitation temperature setpoint signal 702 over a one day period. Thermal mass storage model generator 512 can be configured to generate and provide temperature setpoint signal 702 to smart thermostat 100 during the system identification experiment. Temperature setpoint signal 702 can be generated a priori and tailored to meet the comfort constraints. The minimum and maximum temperature values of temperature setpoint signal 702 may be different depending on whether HVAC equipment 308 are operating in a heating dominated or cooling dominated mode. The values of temperature setpoint signal 702 may define the values of temperature setpoint variable T_(sp) in the input data set used to fit the model parameters.

For variable speed systems, excitation signals generated from a PRBS may not be suitable in practice owing to concerns over saturation effects leading to nonlinearities in the experimental dataset. This issue can be monitored and analyzed via high-fidelity building simulations and setpoint experiments on buildings. Modifications to the excitation signal generation methodology can be evaluated first in simulation to understand the technical trade-offs and limitations. Experiments on buildings can also be performed to help assess the use of other types of excitation signals.

Thermal mass storage model generator 512 can generate a set of input data by modulating the temperature setpoint T_(sp) and recording the values of the temperature setpoint T_(sp) and the outside air temperature T_(oa) at each time step during the system identification period. The set of input data may include a plurality of values of the temperature setpoint T_(sp) and the outside air temperature T_(oa). Thermal mass storage model generator 512 can generate a set of output data by monitoring and recording zone conditions at each time step of the system identification period. The zone conditions may include the zone air temperature T_(ia) and the heating/cooling load {dot over (Q)}_(HVAC). The set of output data may include a plurality of values of the zone air temperature T_(ia) and the heating/cooling load {dot over (Q)}_(HVAC).

The second stage of the system identification process is generating a parameterized building thermal model. The parameterized building thermal model may describe the relationship between zone temperature T_(ia) and sensible HVAC load {dot over (Q)}_(HVAC). In some embodiments, the parameterized building thermal model is based on a thermal circuit representation of building zone 310. An example a thermal circuit representing the heat transfer characteristics of building zone 310 is shown in FIG. 8.

Referring now to FIG. 8, a thermal circuit 800 representing the heat transfer characteristics of building zone 310 is shown, according to some embodiments. Thermal circuit 800 is a two thermal resistance, two thermal capacitance (2R2C) control-oriented thermal mass model. The thermal energy balance within thermal circuit 800 is given by the following linear differential equations:

$\begin{matrix} {{{C_{ia}{\overset{.}{T}}_{ia}} = {{\frac{1}{R_{mi}}\left( {T_{m} - T_{ia}} \right)} + {\frac{1}{R_{oi}}\left( {T_{oa} - T_{ia}} \right)} + {\overset{.}{Q}}_{HVAC} + {\overset{.}{Q}}_{other}}}\mspace{79mu}{{C_{m}{\overset{.}{T}}_{m}} = {\frac{1}{R_{mi}}\left( {T_{ia} - T_{m}} \right)}}} & \left( {{Equation}\mspace{14mu} 13} \right) \end{matrix}$ where T_(ia) represents the indoor air temperature, T_(oa) represents the outdoor air temperature, T_(m) represents the thermal mass temperature (i.e., the average temperature of solid objects in building zone 310), C_(m) represents thermal capacitance of the thermal mass, C_(ia) represents the indoor air thermal capacitance, R_(mi) represents the thermal resistance between the indoor air and the thermal mass, R_(oi) represents the thermal resistance between the indoor air and the outdoor air, {dot over (Q)}_(HVAC) represents the sensible heat added to building zone 310 by HVAC equipment 308 (or removed from building zone 310 if cooling is provided), and {dot over (Q)}_(other) represents the heat load disturbance (e.g., internal heat generation within building zone 310 via solar radiation, occupancy, electrical equipment, etc.).

In some embodiments, heat transfer between adjacent zones is modeled as part of the heat load disturbance {dot over (Q)}_(other) and thus, not explicitly accounted for in the model. In other embodiments, thermal interactions between adjacent zones can be accounted for by modeling the temperature of adjacent zones as additional temperature nodes and adding a thermal resistor between each adjacent zone and building zone 310. For buildings that have significant thermal coupling between zones (e.g., large stores, open floorplan buildings, etc.), it may be desirable to model the thermal interactions between adjacent zones.

The thermal dynamic model of Equation 13 describes the heat transfer between the indoor air and the thermal mass, the heat transfer through the building walls, the sensible heat/cooling duty of HVAC equipment 308 {dot over (Q)}_(HVAC), and the internal heat load {dot over (Q)}_(other) on the space generated by the occupants, electrical plug and lighting, and solar irradiation. Throughout this disclosure, the model of Equation 13 is referred to as the thermal model. The thermal model has been demonstrated to yield acceptable prediction accuracy while managing the trade-off between model complexity (e.g., number of model parameters) and model prediction accuracy. In some embodiments, the internal heat load {dot over (Q)}_(other) cannot be measured and is therefore treated as an unmeasured, time-varying disturbance.

HVAC Load Model Generator

HVAC load model generator 514 can be configured to generate a HVAC load model that describes the relationship between indoor air temperature T_(ia), the temperature setpoint T_(sp), and the sensible HVAC load {dot over (Q)}_(HVAC). In some embodiments, the HVAC load model describes the dynamics of HVAC equipment 308 when providing heating or cooling to building zone 310. For example, the HVAC load model may represent HVAC equipment 308 as a proportional-integral (PI) control system that provides the sensible HVAC load {dot over (Q)}_(HVAC) as a function of the indoor air temperature T_(ia) and the temperature setpoint T_(sp).

In some embodiments, the sensible load {dot over (Q)}_(HVAC) provided by HVAC equipment 308 is a function of the temperature setpoint T_(sp) as well as other factors including the heat gained/lost through the building walls, the internal heat load {dot over (Q)}_(other), and the heat transferred to/removed from the building thermal mass. To provide or remove heat to a particular zone, various actions may be performed by HVAC equipment 308. For example, for a chilled water cooled building with a central air handling unit (AHU) serving several variable air volume (VAV) terminal boxes, a chiller may generate chilled water. The chilled water can be pumped to the AHU cooling coil and air can be forced over the cooling coils of the AHU to cool the air to a supply air temperature setpoint (e.g., 55° F.). The cooled air can be delivered through the supply air duct to terminal VAV boxes at the location of building zone 310. The flow rate of cooled air can be adjusted via a damper such that building zone 310 reaches and/or stays at the temperature setpoint T_(sp).

In some embodiments, the dynamics of the entire process of providing cooling or heating to building zone 310 is non-negligible on the time-scale of interest for utilizing the energy storage of a building mass. To account for the dynamics of the heating/cooling process, HVAC load model generator 514 can model the dynamics between the temperature setpoint T_(sp) and the HVAC sensible load {dot over (Q)}_(HVAC). An example of a model that can be model that can be generated by HVAC load model generator 514 is the following proportional-integral (PI) controller model: {dot over (Q)} _(HVAC,j) =K _(p,j)ε_(sp) +K _(I,j)∫₀ ^(t)ε_(sp)(s)ds ε_(sp) =T _(sp,j) −T _(ia)  (Equation 14) where j∈{clg, hlg} is the index that is used to denote either heating or cooling mode. In some embodiments, different sets of parameters are generated for the heating and cooling modes. In other words, HVAC load model generator 514 can generate a first set of HVAC load model parameters for the cooling mode and a second set of HVAC load model parameters for the heating mode. In some embodiments, the heating and cooling load is constrained to the following set: {dot over (Q)}_(HVAC,clg)∈[−{dot over (Q)}_(clg,cap), 0] and {dot over (Q)}_(HVAC,htg)∈[0, {dot over (Q)}_(htg,cap)] where {dot over (Q)}_(HVAC)<0 indicates that HVAC equipment 308 are providing cooling and {dot over (Q)}_(HVAC)>0 indicates that HVAC equipment 308 are providing heating

The model of Equation 14 includes a proportional gain term (i.e., K_(p,j)ε_(sp)) that accounts for the error ε_(sp) between the indoor air temperature T_(ia) and the setpoint T_(sp,j). The model of Equation 14 also includes an integrator term (i.e., K_(I,j)∫₀ ^(t)ε_(sp)(S) ds) to account for the time-varying nature of the HVAC load {dot over (Q)}_(HVAC,j) required to force the indoor temperature T_(ia) to its setpoint T_(sp,j). The integrator value may be correlated with the internal heat load disturbance {dot over (Q)}_(other) and thus, may play the role of an integrating disturbance model. In some embodiments, the model of Equation 14 does not represent a particular PI controller, but rather is a lumped dynamic model describing the relationship between the HVAC load {dot over (Q)}_(HVAC,j) and the temperature setpoint T_(sp,j). In some embodiments, the HVAC load model defines the HVAC load {dot over (Q)}_(HVAC,j) as a function of the zone air temperature T_(ia), the temperature setpoint T_(sp,j), the zone air humidity H_(ia), a zone air humidity setpoint H_(sp,j), zone heat load {dot over (Q)}_(other), and/or any of a variety of factors that affect the amount of heating or cooling provided by HVAC equipment. It is contemplated that the HVAC load model can include any combination of these or other variables in various embodiments.

In some embodiments, HVAC load model generator 514 accounts for staged HVAC equipment in the HVAC load model. For staged equipment, the sensible heating and cooling load {dot over (Q)}_(HVAC,j) can be modeled to take values in the countable set {dot over (Q)}_(HVAC,j)={{dot over (Q)}_(s1), {dot over (Q)}_(s2), . . . {dot over (Q)}_(sn)}, where {dot over (Q)}_(si)i∈{1, . . . , n} is the HVAC load with the first through ith stage on. The staging down and staging up dynamics can be assumed to be negligible. Alternatively, the staging down and staging up dynamics can be captured using a low-order linear model.

The model of Equation 14 allows for continuous values of {dot over (Q)}_(HVAC,j) and may not be directly applicable to staged HVAC equipment. However, HVAC load model generator 514 can compute the time-averaged HVAC load {dot over (Q)}_(HVAC,j) over a time period, which gives a continuous version of the HVAC load {dot over (Q)}_(HVAC,j) needed to meet the temperature setpoint T_(sp,j). For example, HVAC load model generator 514 can apply a filtering technique to the equipment stage on/off signal to compute a time-averaged version of the HVAC load {dot over (Q)}_(HVAC,j). The filtered version of the HVAC load {dot over (Q)}_(HVAC,j) can then be used as an input to the model of Equation 14 to describe the relationship between the HVAC load {dot over (Q)}_(HVAC,j) and the temperature setpoint T_(sp,j). Throughout this disclosure, the variable {dot over (Q)}_(HVAC,j) is used to represent both the continuous HVAC load when HVAC equipment 308 have a variable speed and to represent the filtered HVAC load when HVAC equipment 308 are staged.

Incorporating the thermal and the HVAC load models together and writing the system of equations as a linear system of differential equations gives the following state space representation:

$\begin{bmatrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{T}}_{m} \\ \overset{.}{I} \end{bmatrix} = {\begin{bmatrix} {\frac{1}{C_{ia}}\left( {K_{p,j} - \frac{1}{R_{mi}} - \frac{1}{R_{oi}}} \right)} & \frac{1}{C_{ia}R_{mi}} & {- \frac{K_{p,j}K_{I,j}}{C_{ia}}} \\ \frac{1}{C_{m}R_{mi}} & {- \frac{1}{C_{m}R_{mi}}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}{\quad{{\begin{bmatrix} T_{ia} \\ T_{m} \\ I \end{bmatrix} + {\begin{bmatrix} {- \frac{K_{p,j}}{C_{ia}}} & \frac{1}{C_{ia}R_{oi}} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} T_{{sp},j} \\ T_{oa} \end{bmatrix}} + {\begin{bmatrix} \frac{1}{C_{ia}} \\ 0 \\ 0 \end{bmatrix}{{\overset{.}{Q}}_{other}\mspace{79mu}\begin{bmatrix} T_{ia} \\ {\overset{.}{Q}}_{{HVAC},j} \end{bmatrix}}}} = {\begin{bmatrix} 1 & 0 & 0 \\ {- K_{p,j}} & 0 & K_{I,j} \end{bmatrix}{\quad{\begin{bmatrix} T_{ia} \\ T_{m} \\ I \end{bmatrix} + {\begin{bmatrix} 0 & 0 \\ K_{p,j} & 0 \end{bmatrix}\begin{bmatrix} T_{{sp},j} \\ T_{oa} \end{bmatrix}}}}}}}}$ or more compactly: {dot over (x)}=A _(c)(θ)x+B _(c)(θ)u+B _(d) d y=C _(c)(θ)x+D _(c)(θ)u  (Equation 15) where x^(T)=[T_(ia), T_(m), I], u^(T)=[T_(sp,j), T_(oa)], d={dot over (Q)}_(other)/C_(ia), y^(T)=[T_(ia), {dot over (Q)}_(HVAC,j)], θ is a parameter vector containing all non-zero entries of the system matrices, and

${{A_{c}(\theta)} = \begin{bmatrix} {- {\theta_{5}\left( {\theta_{1} + \theta_{2} + \theta_{4}} \right)}} & {\theta_{5}\theta_{2}} & {\theta_{3}\theta_{4}\theta_{5}} \\ {\theta_{6}\theta_{2}} & {{- \theta_{6}}\theta_{2}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}},{{B_{c}(\theta)} = \begin{bmatrix} {\theta_{1}\theta_{5}} & {\theta_{4}\theta_{5}} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}},{B_{d} = \begin{bmatrix} \theta_{5} \\ 0 \\ 0 \end{bmatrix}},{{C_{c}(\theta)} = \begin{bmatrix} 1 & 0 & 0 \\ {- \theta_{4}} & 0 & {\theta_{3}\theta_{4}} \end{bmatrix}},{{C_{d}(\theta)} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}},{{D_{c}(\theta)} = \begin{bmatrix} 0 & 0 \\ \theta_{4} & 0 \end{bmatrix}}$

The third stage of the system identification process includes fitting the parameters θ of the parameterized building thermal model (i.e., the model of Equation 15) to the input-output data using a parameter identification algorithm. FIG. 9 is a flowchart of a process 900 which can be performed by system identifier 510 to fit the parameters θ of the parameterized building thermal model. If HVAC equipment 308 is staged, system identifier 510 can filter input-output data to remove the high frequency dither in the indoor air temperature T_(ia) and can compute a time-averaged version of the HVAC load {dot over (Q)}_(HVAC) from the discrete HVAC staging trajectory (step 902). System identifier 510 can then fit the thermal model parameters and HVAC load model parameters θ to the input-output data (or filtered input-output data) obtained during the first stage of the system identification process (step 904). System identifier 510 can augment the resulting state-space model with another integrating disturbance model and can estimate the Kalman filter gain for the resulting model (step 906). Using data obtained in a secondary experiment or under normal operation, system identifier 510 can validate the model through the use of statistics that capture the multi-step prediction accuracy of the resulting model (step 908). Each of these steps is described in detail below.

Step 902 may include filtering the input-output data to remove a high frequency dither in the indoor air temperature T_(ia) and computing a time-averaged version of the HVAC load {dot over (Q)}_(HVAC) from the discrete HVAC staging trajectory. When HVAC equipment 308 are staged, the indoor air temperature trajectory may have a high frequency dither caused by HVAC equipment 308 staging up and down. For example, when operating in a cooling mode, HVAC equipment 308 can be staged up (e.g., by activating discrete chillers) to drive the indoor air temperature T_(ia) to the temperature setpoint T_(sp) minus a deadband. When the indoor air temperature T_(ia) reaches the temperature setpoint T_(sp) minus the deadband, HVAC stages can be deactivated, which results in an increase in the temperature T_(ia). The indoor air temperature T_(ia) may increase until it reaches the temperature setpoint T_(sp) plus the deadband, at which time the HVAC stages can be activated again. The observed effect may be a high frequency dither or oscillation of the temperature T_(ia) around the setpoint T_(sp) (shown in FIG. 10). Also, the HVAC load trajectory may take discrete values as stages are turned on and off. The duty cycle of the on/off trajectory of the stages may depend on the heat transfer to/from building zone 310, the error between the setpoint and indoor air temperature, and the setpoint. Thus, a good indication of the HVAC load {dot over (Q)}_(HVAC) on a continuous scale is some time-average of the HVAC load trajectory or stage on/off trajectories.

To remove the high frequency dither and to perform the time-averaging on the HVAC load trajectory, system identifier 510 can filter the input-output data obtained from the SI experiment. In some embodiments, system identifier 510 uses a first-order Savitzky-Golay filter (SGF) to filter the input-output data. The SGF involves fitting a polynomial to a set of data samples and evaluating the resulting polynomial at a single point within the approximation interval. The SGF is equivalent to performing discrete convolution with a fixed impulse response. However, the SGF is a non-casual filter. To address this issue, system identifier 510 can use the nearest past filtered measurements in the system identification process. For example, let Δt_(filter) be the filtering window of the SGF. For the filter to compute its smoothed estimate of the data at a given time step t, it can use data from t−Δt_(filter)/2 to t+Δt_(filter)/2. To provide the filtered measurements (indoor air temperature T_(ia) and HVAC load {dot over (Q)}_(HVAC)) at a sample time t_(k), the filtered measurement from t_(k)−Δt_(filter)/2 can be used.

FIG. 10 is a pair of graphs 1000 and 1050 illustrating an example of the indoor air temperature T_(ia) trajectory and the HVAC load {dot over (Q)}_(HVAC) trajectory before filtering with the SGF. Line 1002 represents the outdoor air temperature T_(oa) and line 1004 represents the unfiltered indoor air temperature T_(ia). In steady-state, the indoor air temperature T_(ia) may vary around the temperature setpoint within a region that is plus and minus the control deadband 1106. Line 1052 represents the unfiltered sensible load {dot over (Q)}_(HVAC) for staged HVAC equipment.

FIG. 11 is a pair of graphs 1100 and 1150 illustrating an example of the indoor air temperature T_(ia) trajectory and the HVAC load {dot over (Q)}_(HVAC) trajectory after filtering with the SGF. Line 1102 represents the outdoor air temperature T_(oa) and line 1004 represents the filtered indoor air temperature T_(ia). In some embodiments, the filtering removes the variation of temperature about the setpoint to smooth the temperature trajectory. Line 1152 represents the filtered sensible load {dot over (Q)}_(HVAC) for staged HVAC equipment and may be a time-averaged version of the unfiltered sensible load 1052 {dot over (Q)}_(HVAC) shown in FIG. 10.

Referring again to FIG. 9, step 904 may include fitting the thermal model parameters and HVAC load model parameters to the input-output data (or filtered input-output data) obtained during the first stage of the system identification process. In step 904, the time-varying heat load disturbance d may be omitted. System identifier 510 can identify the parameters for the model given by:

$\begin{matrix} {{{\overset{.}{x}(t)} = {{\begin{bmatrix} \theta_{1} & \theta_{2} & \theta_{3} \\ \theta_{4} & {- \theta_{4}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}{x(t)}} + {\begin{bmatrix} {\theta_{5}\theta_{7}} & {\theta_{5}\theta_{6}} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}{u(t)}}}}{{y(t)} = {{\begin{bmatrix} 1 & 0 & 0 \\ {- \theta_{7}} & 0 & \theta_{8} \end{bmatrix}{x(t)}} + {\begin{bmatrix} 0 & 0 \\ \theta_{7} & 0 \end{bmatrix}{u(t)}}}}} & \left( {{Equation}\mspace{14mu} 16} \right) \end{matrix}$ where the effect of the time-varying disturbance d is accounted for via the integrator model.

Using the prediction error method (PEM) described previously, system identifier 510 can obtain an estimate of the model parameters θ. In some embodiments, system identifier 510 estimates the model parameters θ using the functions provided by the Matlab System Identification toolbox. For example, the identification can be performed using the function greyest with the initial state option set to estimate, the disturbance model option set to none, and the estimation focus option is set to simulation. Other options may not be specifically set and therefore default values can be used.

The initial condition of the model of Equation 16 may be unknown, but can be from the data. The disturbance component (i.e., the Kalman gain estimate K) may not be included in the model of Equation 16. The estimation focus can be set to prediction so that the algorithm automatically computes the weighting function of the prediction errors. The stability estimation focus option may perform the same weighting as the prediction option, but also enforces model stability. Under the parameterization of the model of Equation 16, the values of the resistance, capacitance, and PI parameters may be computed from the parameter vector θ if necessary. Given that the resulting PEM optimization problem is a nonlinear, non-convex problem, the initial guess on the parameter values may have a significant impact on the parameter values obtained, and multiple executions of the PEM solver provided a different initial guess each time may converge to different local minima.

To perform step 904 autonomously, model verification can be performed before process 900 continues to 906. If any of the verification steps are not satisfied, the PEM solver can be reinitialized with a different random initial guess. The model verification steps that can be performed in step 904 may include ensuring stability by checking the eigenvalues of the obtained A. In some embodiments, the model verification steps include ensuring that the identified parameters are greater than zero. This is a condition arising from the physical meaningfulness of the model. The model verification steps may include ensuring that the thermal capacitance of the air C_(ia), is less than the thermal capacitance C_(m) of the building mass. In some embodiments, the model verification steps include verifying that the matrix A is well conditioned in order to avoid prediction problems and model sensitivity to noise. If the condition number of A is less than 10500, the conditioning of A may be deemed to be acceptable.

Once the obtained model satisfies all the checks, process 900 may proceed to step 906. It is possible that the conditions described above are not ever satisfied. However, this could be due to a number of reasons that may be difficult to autonomously diagnose and potentially, be associated with a problem beyond the limitations of the SI procedure. To address this problem, a limit can be set for the number of times through the steps above. If the parameter acceptability criteria is not satisfied after the maximum number of iterations through step 904, system identifier 510 can report an error for a human operator to investigate and address.

Estimator Gain Identifier

As discussed above, the model of Equation 16 may only account for the main thermal dynamics of the system and may not identify an appropriate estimator gain. In step 906, the model of Equation 16 can be augmented with another integrating disturbance model and the estimator gain can be identified. In some embodiments, step 906 is performed by estimator gain identifier 516. Estimator gain identifier 516 can be configured to augment the model of Equation 16 to produce the augmented model:

$\begin{bmatrix} {\overset{\overset{.}{\hat{}}}{x}(t)} \\ {\overset{\overset{.}{\hat{}}}{d}(t)} \end{bmatrix} = {{\begin{bmatrix} A_{c} & B_{d} \\ 0 & 0 \end{bmatrix}\begin{bmatrix} {\hat{x}(t)} \\ {\hat{d}(t)} \end{bmatrix}} + {\begin{bmatrix} B_{c} \\ 0 \end{bmatrix}{u(t)}} + {\underset{\underset{= {:{K{(\phi)}}}}{︸}}{\begin{bmatrix} {K_{x}(\phi)} \\ {K_{d}(\phi)} \end{bmatrix}}\left( {{y(t)} - {\hat{y}(t)}} \right)}}$ ${\hat{y}(t)} = {{\left\lbrack {C_{c}\mspace{14mu} C_{d}} \right\rbrack\begin{bmatrix} {\hat{x}(t)} \\ {\hat{d}(t)} \end{bmatrix}} + {D_{c}{u(t)}}}$ where the parameters of A_(c), B_(c), C_(c) and D_(c) are the same as the parameters θ determined in step 904. The disturbance model can be selected such that B_(d)=1 and C_(d)=0, which is an input integrating disturbance model. K(ϕ) is the identified estimator gain parameterized with the parameter vector ϕ as shown in the following equation:

$\begin{matrix} {{{K_{x}(\phi)} = \begin{bmatrix} \phi_{1} & \phi_{2} \\ 0 & 0 \\ \phi_{3} & \phi_{4} \end{bmatrix}},{{K_{d}(\phi)} = \left\lbrack {\phi_{5}\mspace{14mu}\phi_{6}} \right\rbrack}} & \left( {{Equation}\mspace{14mu} 17} \right) \end{matrix}$ where the elements corresponding to the mass temperature T_(m) (i.e., the middle row of the matrix K_(x)(ϕ)) are set to zero. If these elements are allowed to take non-zero values, poor estimation performance may result. Accordingly, estimator gain identifier 516 may set the elements corresponding to the mass temperature T_(m) to zero. The values for the parameters ϕ can be obtained using the function greyest on the augmented model, using the same input-output data as that used in step 904.

Estimator gain identifier 516 can be configured to generate the following augmented matrices:

${A_{aug} = \begin{bmatrix} A_{c} & B_{d} \\ 0 & 0 \end{bmatrix}},{B_{aug} = \begin{bmatrix} B_{c} \\ 0 \end{bmatrix}},{C_{aug} = \left\lbrack {C_{c}\mspace{14mu} C_{d}} \right\rbrack},{D_{aug} = D_{c}}$ Similar to the verifications performed in step 904, estimator gain identifier 516 can check for the stability of the observer system A_(aug)−KC_(aug), and its conditioning number. In this case, the conditioning number of A_(aug)−KC_(aug) may be deemed acceptable if it is less than 70,000.

Step 908 may include validating the augmented model generated in step 906. As discussed above, system identifier 510 may select the optimal model parameters that fit the input-output data with respect to an objective function that depends on the one-step error prediction. In step 904, the PEM is the one-step ahead prediction without estimation (i.e., open-loop estimation), whereas step 906 represents the one-step ahead closed-loop estimation problem. However, within the context of MPC, multi-step open-loop predictions can be used to select the optimal input trajectory over the prediction horizon. Thus, the model validation performed in step 908 may ensure that the estimator is tuned appropriately and that the multi-step open-loop prediction performed with the identified model provides sufficient accuracy.

In some embodiments, system identifier 510 collects another input-output data set for model validation. This data may be collected during normal operation or from another SI experiment. Using this validation data set, system identifier 510 can generate one or more metrics that quantify the multi-step prediction accuracy. One example of a metric which can be generated in step 908 is the Coefficient of Variation Weighted Mean Absolute Prediction Error (CVWMAPE). CVWMAPE is an exponentially weighted average of the absolute prediction error at each time step that quantities the prediction error over time. System identifier 510 can calculate the CVWMAPE as follows:

$\begin{matrix} {{{{CVWMAPE}(k)} = \frac{\sum\limits_{i = k}^{k + N_{h} - 1}{e^{{- i}/N_{h}}{{{y(i)} - {\hat{y}\left( {i❘k} \right)}}}}}{\sum\limits_{i = k}^{k + N_{h} - 1}{e^{{- i}/N_{h}}{{y(i)}}}}},{k = 0},1,2,\ldots} & \left( {{Equation}\mspace{14mu} 18} \right) \end{matrix}$ where N_(h)∈

_(>0) is the prediction horizon, y(i) is the measured output at time step i, and ŷ(i|k) is the predicted output with the identified model given a measurement at time step k and the input sequence u(k), u(k+1), . . . , u(i−1). For notational simplicity the variable y is used to denote one of the two outputs (i.e., in this subsection, y is a scalar). The CVWMAPE can be computed for both outputs.

Another prediction error to consider is prediction error with respect to the q-step ahead prediction (q∈

_(≥0)). System identifier 510 can calculate the Coefficient of Variation Root Mean Squared Prediction Error (CVRMSPE) is to evaluate the q-step ahead prediction error. For example, system identifier 510 can calculate the CVRMSPE for a range of values of q from zero-step ahead prediction up to N_(h)-step ahead prediction. Given a set of measured output values {y(0), . . . , y(M)} for M∈

_(≥0), the CVRMSPE is given by:

$\begin{matrix} {{{{CVRMSPE}(q)} = \frac{{RMSPE}(q)}{\overset{\_}{y}(q)}}{where}} & \left( {{Equation}\mspace{14mu} 19} \right) \\ {{{{RMSPE}(q)} = \sqrt{\frac{\sum\limits_{i = {q + 1}}^{M}\left( {{y(i)} - {\hat{y}\left( {i❘{i - q}} \right)}} \right)}{M - q}}},} & \left( {{Equation}\mspace{14mu} 20} \right) \\ {{{\overset{\_}{y}(q)} = \frac{\sum\limits_{i = q}^{M}{y(i)}}{M - q}},} & \left( {{Equation}\mspace{14mu} 21} \right) \end{matrix}$ for all q∈{0, . . . , N_(h)−1}. The CVRMSPE helps identify the prediction error over the duration of the optimization period.

Once the system identification process is complete, system identifier 510 can provide the system matrices A, B, C, and D with identified parameters θ and the estimator gain K with identified parameters ϕ to state disturbance estimator 520 and predictive optimizer 522. State disturbance estimator 520 can use the system matrices A, B, C, and D and the estimator gain K to compute an estimate of the current state {circumflex over (x)}(k) and unmeasured disturbance {circumflex over (d)}(k) at each sample time.

Load/Rate Predictor

Referring again to FIG. 5, model predictive controller 302 is shown to include a load/rate predictor 518. Load/rate predictor 518 can be configured to predict the heat load disturbance {dot over (Q)}_(other)(k) and the utility rates {circumflex over (r)}_(elec)(k) for each time step k (e.g., k=1 . . . n) of an optimization period. Load/rate predictor 518 can provide the load and rate predictions to predictive optimizer 522 for use in determining the optimal temperature setpoints T_(sp).

In some embodiments, load/rate predictor 518 predicts the value of {dot over (Q)}_(other)(k) at each time step k using a history of disturbance estimates {circumflex over (d)}_(hist) generated by state/disturbance estimator 520. As described above, the state/disturbance estimation problem is given by:

$\begin{matrix} {\begin{bmatrix} {{\hat{T}}_{ia}\left( {{k + 1}❘k} \right)} \\ {{\hat{T}}_{m}\left( {{k + 1}❘k} \right)} \\ {\hat{I}\left( {{k + 1}❘k} \right)} \\ {\hat{d}\left( {{k + 1}❘k} \right)} \end{bmatrix} = {{{A_{aug}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\hat{T}}_{m}\left( {k❘{k - 1}} \right)} \\ {\hat{I}\left( {k❘{k - 1}} \right)} \\ {\hat{d}\left( {k❘{k - 1}} \right)} \end{bmatrix}} + {B_{aug}\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}} + {\begin{bmatrix} K_{x} \\ K_{d} \end{bmatrix}{\left( {\begin{bmatrix} {T_{ia}(k)} \\ {{\overset{.}{Q}}_{HVAC}(k)} \end{bmatrix} - \begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{\overset{.}{\hat{}}}{Q}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}} \right)\mspace{34mu}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{\overset{.}{\hat{}}}{Q}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}}}} = {{C_{aug}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\hat{T}}_{m}\left( {k❘{k - 1}} \right)} \\ {\hat{I}\left( {k❘{k - 1}} \right)} \\ {\hat{d}\left( {k❘{k - 1}} \right)} \end{bmatrix}} + {D\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}}}} & \left( {{Equation}\mspace{14mu} 22} \right) \end{matrix}$ where

${A_{aug} = \begin{bmatrix} A & B_{d} \\ 0 & I \end{bmatrix}},{B_{aug} = \begin{bmatrix} B \\ 0 \end{bmatrix}},{C_{aug} = {\left\lbrack {C\mspace{14mu} 0} \right\rbrack.}}$ State/disturbance estimator 520 can perform the state/disturbance estimation for each time step k to generate a disturbance estimate {circumflex over (d)}(k) for each time step k. State/disturbance estimator 520 can record a history of disturbance estimates

d̂_(hist) := {d̂(k❘k − 1)}_(k = 1)^(N_(hist)) and provide the history of disturbance estimates {circumflex over (d)}_(hist) to load/rate predictor 518.

Load/rate predictor 518 can use the history of disturbance estimates {circumflex over (d)}_(hist) to predict the value of {circumflex over ({dot over (Q)})}_(other)(k) at each time step k. In some embodiments, the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) is a function of the time of day, the day type (e.g., weekend, weekday, holiday), the weather forecasts, building occupancy, and/or other factors which can be used by load/rate predictor 518 to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k). For example, load/rate predictor 518 can predict the heat load disturbance using the equation: {circumflex over ({dot over (Q)})} _(other)(k)=f _(pred)(kΔ+t ₀ ,{circumflex over (d)} _(hist))  (Equation 23) where Δ>0 is the sample period and t₀ is an initial time. In some embodiments, load/rate predictor 518 uses weather forecasts from a weather service 526 and/or load history data from historical data 528 to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) at each time step.

In some embodiments, load/rate predictor 518 uses a deterministic plus stochastic model trained from historical load data to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k). Load/rate predictor 518 may use any of a variety of prediction methods to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). In some embodiments, load/rate predictor 518 makes load/rate predictions using the techniques described in U.S. patent application Ser. No. 14/717,593.

Load/rate predictor 518 is shown receiving utility rates from utilities 524. Utility rates may indicate a cost or price per unit of a resource (e.g., electricity, natural gas, water, etc.) provided by utilities 524 at each time step k in the prediction window. In some embodiments, the utility rates are time-variable rates. For example, the price of electricity may be higher at certain times of day or days of the week (e.g., during high demand periods) and lower at other times of day or days of the week (e.g., during low demand periods). The utility rates may define various time periods and a cost per unit of a resource during each time period. Utility rates may be actual rates received from utilities 524 or predicted utility rates estimated by load/rate predictor 518.

In some embodiments, the utility rates include demand charges for one or more resources provided by utilities 524. A demand charge may define a separate cost imposed by utilities 524 based on the maximum usage of a particular resource (e.g., maximum energy consumption) during a demand charge period. The utility rates may define various demand charge periods and one or more demand charges associated with each demand charge period. In some instances, demand charge periods may overlap partially or completely with each other and/or with the prediction window. Predictive optimizer 522 may be configured to account for demand charges in the high level optimization process performed by predictive optimizer 522. Utilities 524 may be defined by time-variable (e.g., hourly) prices, a maximum service level (e.g., a maximum rate of consumption allowed by the physical infrastructure or by contract) and, in the case of electricity, a demand charge or a charge for the peak rate of consumption within a certain period. Load/rate predictor 518 may store the predicted heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) and the utility rates in memory 508 and/or provide the predicted heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) and the utility rates to predictive optimizer 522.

Predictive Optimizer

Still referring to FIG. 5, model predictive controller 302 is shown to include a predictive optimizer 522. Predictive optimizer 522 can manipulate the temperature setpoint T_(sp) when operating in the operational mode to minimize the economic cost of operating HVAC equipment 308 over the duration of the optimization period. To determine the temperature setpoints T_(sp), predictive optimizer 522 can optimize an objective function (i.e., a cost function) that accounts for the cost of operating HVAC equipment 308 over the duration of the optimization period. The costs of operating HVAC equipment 308 can include, for example, the costs of resources consumed by HVAC equipment 308 during operation (e.g., electricity, natural gas, water, etc.), demand charges imposed by an electric utility, peak load contribution charges, equipment degradation/replacement costs, and/or other costs associated with the operation of HVAC equipment 308.

An example of an objective function which can be optimized by predictive optimizer 522 is shown in the following equation:

$\begin{matrix} {\min\limits_{{T_{{sp},}\epsilon_{T}},{\delta\; T_{sp}}}{\sum\limits_{k = 0}^{N - 1}\left( {{{r_{elec}(k)}{{\hat{\ell}}_{elec}(k)}} + {p_{\epsilon_{T}}{\epsilon_{T}(k)}} + {p_{\delta\; T}\delta\;{T_{sp}(k)}}} \right)}} & \left( {{Equation}\mspace{14mu} 24} \right) \end{matrix}$ where r_(elec)(k) is the price of electricity at time step k, {circumflex over (l)}_(elec)(k) is the predicted electricity consumption of HVAC equipment 308 at time step k, ∈_(T)(k) is a number of degrees by which the zone temperature constraints are violated at time step k, p_(∈) _(T) is a zone air temperature penalty coefficient applied to ∈_(T) (k), δT_(sp)(k) is a number of degrees by which the zone temperature setpoint T_(sp) changes between time step k−1 and time step k, and p_(δT) _(sp) is a temperature setpoint change penalty coefficient applied to δT_(sp)(k). Equation 24 accounts for the cost of electricity consumption (r_(elec)(k){circumflex over (l)}_(elec)(k)), penalizes violations of the indoor air temperature bounds (p_(∈) _(T) ∈_(T)(k)), and penalizes changes in the temperature setpoint (p_(δT)δT_(sp)(k)).

If additional resources (other than electricity) are consumed by HVAC equipment 308, additional terms can be added to Equation 24 to represent the cost of each resource consumed by HVAC equipment 308. For example, if HVAC equipment 308 consume natural gas and water in addition to electricity, Equation 24 can be updated to include the terms r_(gas)(k){circumflex over (l)}_(gas)(k) and r_(water)(k){circumflex over (l)}_(water)(k) in the summation, where r_(gas)(k) is the cost per unit of natural gas at time step k, {circumflex over (l)}_(gas)(k) is the predicted natural gas consumption of HVAC equipment 308 at time step k, r_(water) (k) is the cost per unit of water at time step k, and {circumflex over (l)}_(water) (k) is the predicted water consumption of HVAC equipment 308 at time step k.

Predictive optimizer 522 can be configured to estimate the amount of each resource consumed by HVAC equipment 308 (e.g., electricity, natural gas, water, etc.) as a function of the sensible heating or cooling load {circumflex over ({dot over (Q)})}_(HVAC). In some embodiments, a constant efficiency model is used to compute the resource consumption of HVAC equipment 308 as a fixed multiple heating or cooling load {circumflex over ({dot over (Q)})}_(HVAC) (e.g., {circumflex over (l)}_(elec)(k)=η{circumflex over ({dot over (Q)})}_(HVAC)(k)). In other embodiments, equipment models describing a relationship between resource consumption and heating/cooling production can be used to approximate the resource consumption of the HVAC equipment 308. The equipment models may account for variations in equipment efficiency as a function of load and/or other variables such as outdoor weather conditions (e.g., {circumflex over (l)}_(elec)(k)=f({circumflex over ({dot over (Q)})}_(HVAC)(k),T_(oa)(k),η_(base))). Predictive optimizer 522 can convert the estimated heating/cooling load {circumflex over ({dot over (Q)})}_(HVAC)(k) at each time step to one or more resource consumption values (e.g., {circumflex over (l)}_(elec)(k), {circumflex over (l)}_(gas)(k), {circumflex over (l)}_(water)(k), etc.) for inclusion in the objective function.

In some embodiments, predictive optimizer 522 is configured to modify the objective function to account for various other costs associated with operating HVAC equipment 308. For example, predictive optimizer 522 can modify the objective function to account for one or more demand charges, peak load contribution charges, equipment degradation costs, equipment purchase costs, revenue generated from participating in incentive-based demand response programs, economic load demand response, and/or any other factors that can contribute to the cost incurred by operating HVAC equipment 308 or revenue generated by operating HVAC equipment 308. As used herein, the term “energy cost” can include any such charges, costs, revenues, incentives, etc., including combinations thereof, in various embodiments. Predictive optimizer 522 can add one or more additional terms to the objective function to account for these or other factors. Several examples of such functionality are described in U.S. patent application Ser. No. 15/405,236 filed Jan. 12, 2017, U.S. patent application Ser. No. 15/405,234 filed Jan. 12, 2017, U.S. patent application Ser. No. 15/426,962 filed Feb. 7, 2017, U.S. patent application Ser. No. 15/473,496 filed Mar. 29, 2017, and U.S. patent application Ser. No. 15/616,616 filed Jun. 7, 2017. The entire disclosure of each of these patent applications is incorporated by reference herein. In some embodiments, model predictive controller 302 includes some or all of the functionality of the controllers and/or control systems described in these patent applications.

Predictive optimizer 522 can be configured to automatically generate and impose constraints on the optimization of the objective function. The constraints may be based on the system model provided by system identifier 510, the state estimates provided by state/disturbance estimator 520, the heat load disturbance predictions provided by load/rate predictor 518, and constraints on the zone air temperature T_(ia). For example, predictive optimizer 522 can generate and impose the following constraints for each time step k∈{0, . . . , N−1}:

$\begin{matrix} {\begin{bmatrix} {T_{ia}\left( {k + 1} \right)} \\ {T_{m}\left( {k + 1} \right)} \\ {I\left( {k + 1} \right)} \end{bmatrix} = {{A\begin{bmatrix} {T_{ia}(k)} \\ {T_{m}(k)} \\ {I(k)} \end{bmatrix}} + {B\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}} + {B_{d}{{\overset{.}{Q}}_{other}(k)}}}} & \left( {{Equation}\mspace{14mu} 25} \right) \\ {\mspace{79mu}{\left\lbrack {{\overset{.}{Q}}_{HVAC}(k)} \right\rbrack = {{C\begin{bmatrix} {T_{ia}(k)} \\ {T_{m}(k)} \\ {I(k)} \end{bmatrix}} + {D\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}}}} & \left( {{Equation}\mspace{14mu} 26} \right) \end{matrix}$ {circumflex over (l)} _(elec)(k)=η{circumflex over ({dot over (Q)})}_(HVAC)(k)  (Equation 27) T _(ia)(k)≤T _(max)(k)+T _(deadband)+∈_(T)(k)  (Equation 28) T _(ia)(k)≥T _(min)(k)−T _(deadband)−∈_(T)(k)  (Equation 29) T _(sp)(k)∈[T _(sp,min)(k),T _(sp,max)(k)]  (Equation 30) T _(sp)(k+1)−T _(sp)(k)≤δT _(sp)(k)  (Equation 31) −T _(sp)(k+1)+T _(sp)(k)≤δT _(sp)(k)  (Equation 32) ∈_(T)(k)≥0  (Equation 33) T _(ia)(0)={circumflex over (T)} _(ia,0) ,T _(m)(0)={circumflex over (T)} _(m,0) ,I(0)=Î ₀  (Equation 34)

Equation 25 is based on the thermal mass storage model provided by thermal mass storage model generator 512. The thermal mass storage model predicts the system states at each time step k+1 as a function of the system states x(k), inputs u(k), and heat load disturbance at the previous time step k. Specifically, Equation 25 defines the relationship between the system states T_(ia)(k+1), T_(m)(k+1), and I(k+1) at time step k+1, the system states T_(ia)(k), T_(m)(k), and I(k) at time step k, the controlled or measured inputs T_(sp)(k) and T_(oa)(k) at time step k, and the estimated heat load disturbance {dot over (Q)}_(other)(k) at time step k. The parameters in the matrices A and B can be determined by system identifier 510, as previously described. The value of {dot over (Q)}_(other)(k) at each time step k can be determined by load/rate predictor 518.

Equation 26 is based on the HVAC load model provided by HVAC load model generator 514. The HVAC load model defines the heating/cooling load {dot over (Q)}_(HVAC)(k) at each time step k as a function of the system states T_(ia)(k), T_(m)(k), and I(k) at time step k and the controlled or measured inputs T_(sp)(k) and T_(oa)(k) at time step k. The parameters in the matrices C and D can be determined by system identifier 510, as previously described.

Equation 27 defines the relationship between the predicted electric consumption {circumflex over (l)}_(elec)(k) at time step k and the heating/cooling load {dot over (Q)}_(HVAC)(k) at time step k. If additional resources (other than electricity) are consumed by HVAC equipment 308, additional constraints can be added to define the relationship between {dot over (Q)}_(HVAC)(k) and the amount of each resource consumed by HVAC equipment 308.

Equations 28 and 29 constrain the zone air temperature T_(ia)(k) at each time step k between the minimum temperature threshold T_(min)(k) and the maximum temperature threshold T_(max)(k) at that time step k, plus or minus the temperature deadband T_(deadband) and a temperature error ∈_(T)(k). In some embodiments, the values of the minimum temperature threshold T_(min)(k) and the maximum temperature threshold T_(max)(k) can vary over the duration of the optimization period. For example, the minimum temperature threshold T_(min)(k) and the maximum temperature threshold T_(max)(k) may define a first zone temperature range (i.e., T_(min)(k) to T_(max)(k)) during time step k and a second zone temperature range (i.e., T_(min)(k+1) to T_(max)(k+1)) during time step k+1. The values of T_(min) and T_(max) at each time step can vary as a function of the time of day, day of the week, building occupancy, or other factors. In some embodiments, the values of T_(min) and T_(max) are input by a user, can be adjusted by a user, or calculated based on another user input or behavior. Predictive optimizer 522 may be allowed to violate the temperature constraints if necessary, but any deviation from the defined temperature range (i.e., between T_(min)(k)−T_(deadband) and T_(max)(k)+T_(deadband)) may be penalized in the objective function by imposing a penalty p_(∈) _(T) on the temperature error ∈_(T)(k). Equation 33 requires the temperature error ∈_(T)(k) to be non-negative at each time step k.

Equation 30 limits the temperature setpoint T_(sp) (k) at each time step between a minimum temperature setpoint T_(sp,min)(k) and a maximum temperature setpoint T_(sp,max)(k), whereas Equations 31-32 define the changes δT_(sp)(k) in the temperature setpoint T_(sp)(k) between consecutive time steps. The minimum temperature setpoint T_(sp,min)(k) and the maximum temperature setpoint T_(sp,max)(k) may be different from the minimum temperature threshold T_(min)(k) and the maximum temperature threshold T_(max)(k) used to constrain the zone air temperature T_(ia)(k). In some embodiments, the values of the minimum temperature setpoint T_(sp,min)(k) and the maximum temperature setpoint T_(sp,max)(k) can vary over the duration of the optimization period. For example, the minimum temperature setpoint T_(sp,min)(k) and the maximum temperature setpoint T_(sp,max)(k) may define a first temperature setpoint range (i.e., T_(sp,min)(k) to T_(sp,max)(k)) during time step k and a second temperature setpoint range (i.e., T_(sp,min)(k+1) to T_(sp,max)(k+1)) during time step k+1. The values of T_(sp,min) and T_(sp,max) at each time step can vary as a function of the time of day, day of the week, building occupancy, or other factors. In some embodiments, the values of T_(sp,min) and T_(sp,max) are input by a user, can be adjusted by a user, or calculated based on another user input or behavior. Changes in the temperature setpoint T_(sp) may be penalized in the objective function by imposing a penalty p_(δT) _(sp) on the temperature change δT_(sp)(k).

Equation 34 sets the initial system states for the zone air temperature T_(ia)(0), the zone mass temperature T_(m)(0), and the integrated disturbance I(0) at the first time step k=0 of the optimization period. The initial system states T_(ia)(0), T_(m)(0), and I(0) may be set to the initial state estimates {circumflex over (T)}_(ia,0), {circumflex over (T)}_(m,0), and Î₀ provided by state/disturbance estimator 520. In other words, the predictive model shown in Equations 25-26 may be initialized using the state estimates provided by state/disturbance estimator 520.

Predictive optimizer 522 can be configured to optimize the objective function (Equation 24) subject to the optimization constraints (Equations 25-34) to determine optimal values of the temperature setpoint T_(sp) (k) at each time step k of the optimization period. Predictive optimizer 522 can use any of a variety of optimization techniques to perform the optimization. For example, the optimization problem can be formulated as a linear program (Equations 24-34) and solved using a linear optimization technique (e.g., a basis exchange algorithm, an interior point algorithm, a cutting-plane algorithm, a branch and bound algorithm, etc.) using any of a variety solvers and/or programming languages. In some embodiments, predictive optimizer 522 performs the optimization at the beginning of each time step k to determine optimal temperature setpoints T_(sp)(k) for the next N time steps.

Once the optimal temperature setpoints T_(sp) have been determined, predictive optimizer 522 can provide the optimal temperature setpoint T_(sp) (k) for the current time step k to smart thermostat 100 and/or equipment controller 406. State/disturbance estimator 520 can use the temperature setpoint T_(sp) (k) for the current time step k along with feedback received from HVAC equipment 308 and building zone 310 during time step k (e.g., measurements of T_(ia)(k), T_(oa)(k), {dot over (Q)}_(HVAC)(k), etc.) to update the state/disturbance estimates the next time the optimization is performed.

Model Predictive Control Processes

Referring now to FIGS. 12-14, several flowcharts of model predictive control processes 1200-1400 are shown, according to some embodiments. Processes 1200-1400 can be performed by one or more components of model predictive control systems 300-400, as described with reference to FIGS. 3-11. For example, processes 1200-1400 can be performed by model predictive controller 302, smart thermostat 100, equipment controller 406, and/or HVAC equipment 308 to control the temperature of building zone 310.

Referring specifically to FIG. 12, process 1200 is shown to include operating a model predictive controller in system identification mode to identify parameters of a thermal mass storage model and a HVAC load model for a building zone (step 1202). In some embodiments, step 1202 is performed by system identifier 510. Step 1202 can include generating a thermal mass storage model by modeling the heat transfer characteristics of the building zone using a thermal circuit 800, as described with reference to FIG. 8. Step 1202 may also include generating a HVAC load model by modeling the amount of heating/cooling {dot over (Q)}_(HVAC) provided by HVAC equipment 308 as a function of the temperature setpoint T_(sp) and the zone air temperature T_(ia). The thermal mass storage model and the HVAC load model can be used to generate a system of differential equations that define the relationship between zone air temperature T_(ia), zone mass temperature T_(m), outdoor air temperature T_(oa), the amount of heating or cooling {dot over (Q)}_(HVAC) provided by HVAC equipment 308, the heat load disturbance {dot over (Q)}_(other), and the temperature setpoint T_(sp).

The unknown parameters in the thermal mass storage model and the HVAC load model (e.g., R_(oi), R_(mi), C_(m), C_(ia)) can be organized into parameter matrices A, B, C, and D, as shown in the following equations:

$\begin{bmatrix} {\overset{.}{T}}_{ia} \\ \overset{.}{T_{m}} \\ \overset{.}{I} \end{bmatrix} = {{\begin{bmatrix} {\frac{1}{C_{ia}}\left( {K_{p,j} - \frac{1}{R_{m\; i}} - \frac{1}{R_{oi}}} \right)} & \frac{1}{C_{ia}R_{m\; i}} & {- \frac{K_{p,j}K_{l,j}}{C_{ia}}} \\ \frac{1}{C_{m}R_{m\; i}} & {- \frac{1}{C_{m}R_{m\; i}}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}\left\lbrack \begin{matrix} T_{ia} \\ T_{m} \\ I \end{matrix} \right\rbrack} + {\quad{{{\begin{bmatrix} {- \frac{K_{p,j}}{C_{ia}}} & \frac{1}{C_{ia}R_{oi}} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} T_{{sp},j} \\ T_{oa} \end{bmatrix}} + {\begin{bmatrix} \frac{1}{C_{ia}} \\ 0 \\ 0 \end{bmatrix}{{\overset{.}{Q}}_{other}\mspace{20mu}\begin{bmatrix} T_{ia} \\ {\overset{.}{Q}}_{{HVAC},j} \end{bmatrix}}}} = {{\begin{bmatrix} 1 & 0 & 0 \\ {- K_{p,j}} & 0 & K_{I,j} \end{bmatrix}\begin{bmatrix} T_{ia} \\ T_{m} \\ I \end{bmatrix}} + {\begin{bmatrix} 0 & 0 \\ K_{p,j} & 0 \end{bmatrix}\begin{bmatrix} T_{{sp},j} \\ T_{oa} \end{bmatrix}}}}}}$ or more compactly: {dot over (x)}=A _(c)(θ)x+B _(c)(θ)u+B _(d) d y=C _(c)(θ)x+D _(c)(θ)u where x^(T)=[T_(ia), T_(m), I], u^(T)=[T_(sp,j), T_(oa)], d={dot over (Q)}_(other)/C_(ia), y^(T)=[T_(ia), {dot over (Q)}_(HVAC,j)], θ is a parameter vector containing all non-zero entries of the system matrices, and

${{A_{c}(\theta)} = \begin{bmatrix} {- {\theta_{5}\left( {\theta_{1} + \theta_{2} + \theta_{4}} \right)}} & {\theta_{5}\theta_{2}} & {\theta_{3}\theta_{4}\theta_{5}} \\ {\theta_{6}\theta_{2}} & {{- \theta_{6}}\theta_{2}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}},{{B_{c}(\theta)} = \begin{bmatrix} {\theta_{1}\theta_{5}} & {\theta_{4}\theta_{5}} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}},{B_{d} = \begin{bmatrix} \theta_{5} \\ 0 \\ 0 \end{bmatrix}},{{C_{c}(\theta)} = \begin{bmatrix} 1 & 0 & 0 \\ {- \theta_{4}} & 0 & {\theta_{3}\theta_{4}} \end{bmatrix}},{{C_{d}(\theta)} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}},{{D_{c}(\theta)} = \begin{bmatrix} 0 & 0 \\ \theta_{4} & 0 \end{bmatrix}}$

The unknown parameters θ in the thermal mass storage model and the HVAC load model (i.e., the parameters in matrices A, B, C, and D) can be identified by fitting the parameters θ to a set of input-output data (e.g., sets of values T_(ia), T_(oa), T_(sp), and {dot over (Q)}_(HVAC)) collected from the HVAC system when operating in the system identification mode. In some embodiments, the input-output data are collected by modulating the temperature setpoint T_(sp) and observing the corresponding values of T_(ia), T_(oa), and {dot over (Q)}_(HVAC) for a plurality of time steps during the system identification mode. The process of fitting the model parameters θ to the set of input-output data can be accomplished by performing process 900, as described with reference to FIG. 9.

If HVAC equipment 308 are staged, step 1202 can include filtering input-output data to remove the high frequency dither in the indoor air temperature T_(ia) and can compute a time-averaged version of the HVAC load {dot over (Q)}_(HVAC) from the discrete HVAC staging trajectory. The thermal model parameters and HVAC load model parameters θ can then be fit to the input-output data (or filtered input-output data). In some embodiments, step 1202 includes augmenting the resulting state-space model with another integrating disturbance model and estimating the Kalman filter gain for the resulting model. Using data obtained in a secondary experiment or under normal operation, step 1202 can include validating the model through the use of statistics that capture the multi-step prediction accuracy of the resulting model.

Still referring to FIG. 12, process 1200 is shown to include operating the model predictive controller in operational mode to determine temperature setpoints T_(sp) for the building zone (step 1204). In some embodiments, step 1204 is performed by predictive optimizer 522, as described with reference to FIG. 5. To determine the temperature setpoints T_(sp), step 1204 can include optimizing an objective function (i.e., a cost function) that accounts for the cost of operating HVAC equipment 308 over the duration of the optimization period. The costs of operating HVAC equipment 308 can include, for example, the costs of resources consumed by HVAC equipment 308 during operation (e.g., electricity, natural gas, water, etc.), demand charges imposed by an electric utility, peak load contribution charges, equipment degradation/replacement costs, and/or other costs associated with the operation of HVAC equipment 308. The objective function can be optimized subject to constraints provided by the thermal mass storage model and the HVAC load model, as well as constraints on the zone temperature T_(ia). Step 1204 can be accomplished by performing process 1300, described in greater detail with reference to FIG. 13.

In some embodiments, process 1200 includes automatically switching between the system identification mode and the operational mode based on a prediction error of the thermal mass storage model and/or the HVAC load model. For example, process 1200 can include comparing the value of the zone temperature {circumflex over (T)}_(ia)(k|k−1) predicted by the thermal mass storage model to the actual measured value of the zone temperature T_(ia)(k). If the difference between the predicted value {circumflex over (T)}_(ia)(k|k−1) and the actual value T_(ia)(k) exceeds an error threshold, process 1200 may automatically return to step 1202 and repeat the system identification process. Similarly, process 1200 can include comparing the value of the HVAC equipment load {dot over ({circumflex over (Q)})}_(HVAC) (k|k−1) predicted by the HVAC load model to the actual value of the HVAC load {dot over (Q)}_(HVAC) (k). If the difference between the predicted value {dot over ({circumflex over (Q)})}_(HVAC)(k|k−1) and the actual value {dot over (Q)}_(HVAC)(k) exceeds an error threshold, process 1200 may automatically return to step 1202 and repeat the system identification process.

Process 1200 is shown to include providing the optimal temperature setpoints T_(sp) to an equipment controller (step 1206) and using the equipment controller to operate HVAC equipment to drive the temperature of the building zone T_(ia) to the optimal temperature setpoints T_(sp) (step 1208). Step 1206 can include sending the optimal temperature setpoints T_(sp) from model predictive controller 302 to an equipment controller 406. In some embodiments, both model predictive controller 302 and equipment controller 406 are components of a smart thermostat 100 (as shown in FIG. 4). In other embodiments, model predictive controller 302 is separate from smart thermostat 100 and configured to send the optimal temperature setpoints to smart thermostat 100 via a communications network 304 (as shown in FIG. 3). Smart thermostat 100 and/or equipment controller 406 can use the optimal temperature setpoints T_(sp) to generate control signals for HVAC equipment 308 which operate to drive the temperature of the building zone T_(ia) to the optimal temperature setpoints T_(sp).

Referring now to FIG. 13, a flowchart of a process 1300 for generating optimal temperature setpoints for a building zone is shown, according to some embodiments. In some embodiments, process 1300 is performed by model predictive controller 302, as described with reference to FIGS. 5-11. Process 1300 can be performed to accomplish step 1204 of process 1200.

Process 1300 is shown to include predicting a heat load disturbance {dot over (Q)}_(other) experienced by a building zone at each time step (k=0 . . . N−1) of an optimization period (step 1302). In some embodiments, step 1302 is performed by load/rate predictor 518 and state/disturbance predictor 520. Step 1302 can include generating a disturbance estimate {circumflex over (d)}(k) for each time step k in the optimization period. In some embodiments, step 1302 includes estimating the disturbance state {circumflex over (d)}(k) at each time step k in the optimization period using the following state/disturbance model:

$\begin{bmatrix} {{\hat{T}}_{ia}\left( {{k + 1}❘k} \right)} \\ {{\hat{T}}_{m}\left( {{k + 1}❘k} \right)} \\ {\hat{I}\left( {{k + 1}❘k} \right)} \\ {\hat{d}\left( {{k + 1}❘k} \right)} \end{bmatrix} = {{{A_{aug}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\hat{T}}_{m}\left( {k❘{k - 1}} \right)} \\ {\hat{I}\left( {k❘{k - 1}} \right)} \\ {\hat{d}\left( {k❘{k - 1}} \right)} \end{bmatrix}} + {B_{aug}\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}} + {\begin{bmatrix} K_{x} \\ K_{d} \end{bmatrix}{\left( {\begin{bmatrix} {T_{ia}(k)} \\ {{\overset{.}{Q}}_{HVAC}(k)} \end{bmatrix} - \begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{\overset{.}{\hat{}}}{Q}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}} \right)\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{\overset{.}{\hat{}}}{Q}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}}}} = {{C_{aug}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\hat{T}}_{m}\left( {k❘{k - 1}} \right)} \\ {\hat{I}\left( {k❘{k - 1}} \right)} \\ {\hat{d}\left( {k❘{k - 1}} \right)} \end{bmatrix}} + {D\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}}}$ where

${A_{aug} = \begin{bmatrix} A & B_{d} \\ 0 & I \end{bmatrix}},{B_{aug} = \begin{bmatrix} B \\ 0 \end{bmatrix}},{C_{aug} = {\left\lbrack {C\mspace{14mu} 0} \right\rbrack.}}$ Step 1302 can include recording a history of disturbance estimates {circumflex over (d)}_(hist):={{circumflex over (d)}(k|k−1)}_(k=1) ^(N) ^(hist) predicted by the state/disturbance model.

Step 1302 can include using the history of disturbance estimates {circumflex over (d)}_(hist) to predict the value of {circumflex over ({dot over (Q)})}_(other)(k) at each time step k. In some embodiments, the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) is a function of the time of day, the day type (e.g., weekend, weekday, holiday), the weather forecasts, building occupancy, and/or other factors which can be used by load/rate predictor 518 to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k). For example, step 1302 can include predicting the heat load disturbance using the equation: {circumflex over ({dot over (Q)})}_(other)(k)=f _(pred)(kΔ+t ₀ ,{circumflex over (d)} _(hist)) where Δ>0 is the sample period and t₀ is an initial time. In some embodiments, step 1302 includes using weather forecasts from a weather service 526 and/or load history data from historical data 528 to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) at each time step.

In some embodiments, step 1302 includes using a deterministic plus stochastic model trained from historical load data to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k). Step 1302 can include using any of a variety of prediction methods to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). In some embodiments, step 1302 includes predicting the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) using the techniques described in U.S. patent application Ser. No. 14/717,593.

Still referring to FIG. 13, process 1300 is shown to include estimating initial system states {circumflex over (x)}(0) for a first time step (k=0) of the optimization period (step 1304). The initial system states {circumflex over (x)}(0) may include the estimated zone air temperature {circumflex over (T)}_(ia)(0), the estimated zone mass temperature {circumflex over (T)}_(m)(0), and the estimated integrating disturbance Î(0) at the beginning of the optimization period. In some embodiments, step 1304 is performed by state/disturbance estimator 520. The initial system states {circumflex over (x)}(0) can be estimated using same the state/disturbance estimation model used in step 1302. For example, step 1304 can include estimating the system states {circumflex over (x)}(k) at each time step k using the following equation: {circumflex over (x)}(k+1|k)=A{circumflex over (x)}(k|k−1)+Bu(k)+K(y(k)−ŷ(k|k−1)) ŷ(k|k−1)=C{circumflex over (x)}(k|k−1)+Du(k) where {circumflex over (x)}(k+1|k) is the estimated/predicted state at time step k+1 given the measurement at time step k, ŷ(k|k−1) is the predicted output at time step k given the measurement at time step k−1, and K is the estimator gain.

In some embodiments, the estimated state vector {circumflex over (x)}(k+1|k), the output vector ŷ(k|k−1), and the input vector u(k) are defined as follows:

${{\hat{x}\left( {{k + 1}❘k} \right)} = \begin{bmatrix} {{\hat{T}}_{ia}\left( {{k + 1}❘k} \right)} \\ {{\hat{T}}_{m}\left( {{k + 1}❘k} \right)} \\ {\hat{I}\left( {{k + 1}❘k} \right)} \end{bmatrix}},{{\hat{y}\left( {k❘{k - 1}} \right)} = \begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{.}{\hat{Q}}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}},{{u(k)} = \begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}$ where {circumflex over (T)}_(ia) is an estimate of the zone air temperature T_(ia), {circumflex over (T)}_(m) is an estimate of the zone mass temperature T_(m), Î is an estimate of the integrating disturbance I, {circumflex over ({dot over (Q)})}_(HVAC) is an estimate of the heating or cooling load provided by HVAC equipment 308, T_(sp) is the temperature setpoint, and T_(oa) is the outdoor air temperature. The estimated system states {circumflex over (x)}(0) at the first time step k=0 can be used as the initial system states.

Still referring to FIG. 13, process 1300 is shown to include using a thermal mass storage model and a heat load model to predict the temperature of the building zone {circumflex over (T)}_(ia) and the HVAC equipment load {circumflex over ({dot over (Q)})}_(HVAC) at each time step as a function of the estimated system states {circumflex over (x)}, temperature setpoint trajectory T_(sp), and estimated heat load disturbance {circumflex over ({dot over (Q)})}_(other) (step 1306). The thermal mass storage model may be defined as follows:

$\begin{bmatrix} {{\hat{T}}_{ia}\left( {k + 1} \right)} \\ {{\hat{T}}_{m}\left( {k + 1} \right)} \\ {\hat{I}\left( {k + 1} \right)} \end{bmatrix} = {{A\begin{bmatrix} {{\hat{T}}_{ia}(k)} \\ {{\hat{T}}_{m}(k)} \\ {\hat{I}(k)} \end{bmatrix}} + {B\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}} + {B_{d}{{\overset{\overset{\bigwedge}{.}}{Q}}_{other}(k)}}}$ where {circumflex over (T)}_(ia)(k+1) is the predicted temperature of the building zone at time step k+1,{circumflex over (T)}_(m)(k+1) is the predicted temperature of the building mass at time step k+1, Î(k+1) is the predicted value of the integrating disturbance at time step k+1, {circumflex over (T)}_(ia)(k) is the predicted temperature of the building zone at time step k, {circumflex over (T)}_(m)(k) is the predicted temperature of the building mass at time step k, Î(k) is the predicted value of the integrating disturbance at time step k, T_(sp)(k) is the temperature setpoint at time step k, T_(oa)(k) is the outdoor air temperature (measured) at time step k, and {dot over ({circumflex over (Q)})}_(other)(k) is the estimated heat load disturbance at time step k.

The HVAC load model may be defined as follows:

$\left\lbrack {{\overset{\overset{\bigwedge}{.}}{Q}}_{HVAC}(k)} \right\rbrack = {{C\begin{bmatrix} {{\hat{T}}_{ia}(k)} \\ {{\hat{T}}_{m}(k)} \\ {\hat{I}(k)} \end{bmatrix}} + {D\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}}$ where {dot over ({circumflex over (Q)})}_(HVAC)(k) is the predicted HVAC equipment load at time step k and {circumflex over (T)}_(ia)(k), {circumflex over (T)}_(m)(k), Î(k), T_(sp)(k), and T_(oa)(k) are the same as the corresponding variables in the thermal mass storage model.

Process 1300 is shown to include optimizing an economic cost function subject to constraints on the predicted temperature of the building zone {circumflex over (T)}_(ia) to determine optimal temperature setpoints T_(sp) for the building zone at each time step of the optimization period (step 1308). An example of an objective function which can be optimized in step 1308 is shown in the following equation:

$\min\limits_{T_{sp},\epsilon_{T},{\delta\; T_{sp}}}{\sum\limits_{k = 0}^{N - 1}\left( {{{r_{elec}(k)}{{\hat{\ell}}_{elec}(k)}} + {p_{\epsilon_{T}}{\epsilon_{T}(k)}} + {p_{\delta\; T}\delta\;{T_{sp}(k)}}} \right)}$ where r_(elec)(k) is the price of electricity at time step k, {circumflex over (l)}_(elec)(k) is the predicted electricity consumption of HVAC equipment 308 at time step k, ∈_(T)(k) is a number of degrees by which the zone temperature constraints are violated at time step k, p_(∈) _(T) is a zone air temperature penalty coefficient applied to ∈_(T) (k), δT_(sp)(k) is a number of degrees by which the zone temperature setpoint T_(sp) changes between time step k−1 and time step k, and p_(δT) _(sp) is a temperature setpoint change penalty coefficient applied to δT_(sp)(k). Equation 24 accounts for the cost of electricity consumption (r_(elec)(k){circumflex over (l)}_(elec)(k)), penalizes violations of the indoor air temperature bounds (p_(∈) _(T) ∈_(T)(k)), and penalizes changes in the temperature setpoint (p_(δ) _(T) δT_(sp)(k)).

If additional resources (other than electricity) are consumed by HVAC equipment 308, additional terms can be added to the objective function to represent the cost of each resource consumed by HVAC equipment 308. For example, if HVAC equipment 308 consume natural gas and water in addition to electricity, the objective function can be updated to include the terms r_(gas)(k){circumflex over (l)}_(gas)(k) and r_(water)(k){circumflex over (l)}_(water)(k) in the summation, where r_(gas)(k) is the cost per unit of natural gas at time step k, {circumflex over (l)}_(gas)(k) is the predicted natural gas consumption of HVAC equipment 308 at time step k, r_(water) (k) is the cost per unit of water at time step k, and {circumflex over (l)}_(water)(k) is the predicted water consumption of HVAC equipment 308 at time step k.

In some embodiments, step 1308 includes estimating the amount of each resource consumed by HVAC equipment 308 (e.g., electricity, natural gas, water, etc.) as a function of the sensible heating or cooling load {circumflex over ({dot over (Q)})}_(HVAC). In some embodiments, a constant efficiency model is used to compute the resource consumption of HVAC equipment 308 as a fixed multiple heating or cooling load {circumflex over ({dot over (Q)})}_(HVAC) (e.g., {circumflex over (l)}_(elec)(k)=η{circumflex over ({dot over (Q)})}_(HVAC)(k)). In other embodiments, equipment models describing a relationship between resource consumption and heating/cooling production can be used to approximate the resource consumption of the HVAC equipment 308. The equipment models may account for variations in equipment efficiency as a function of load and/or other variables such as outdoor weather conditions (e.g., {circumflex over (l)}_(elec)(k)=f({circumflex over ({dot over (Q)})}_(HVAC)(k), T_(oa)(k), η_(base))). Step 1308 can include converting the estimated heating/cooling load {circumflex over ({dot over (Q)})}_(HVAC)(k) at each time step to one or more resource consumption values (e.g., {circumflex over (l)}_(elec)(k), {circumflex over (l)}_(gas)(k), {circumflex over (l)}_(water)(k), etc.) for inclusion in the objective function.

Step 1308 can include optimizing the objective function to determine optimal values of the temperature setpoint T_(sp) (k) for each time step in the optimization period. The objective function can be optimized subject to a set of constraints (e.g., Equations 25-34). The constraints may include the thermal mass storage model and the HVAC load model, which define the relationship between {circumflex over ({dot over (Q)})}_(HVAC), the temperature setpoints T_(sp), and the zone air temperature T_(ia) at each time step. The constraints may also include constraints on the zone air temperature T_(ia)(k) and constraints that define the penalty terms in the objective function.

In some embodiments, step 1308 includes performing the optimization at the beginning of each time step k to determine optimal temperature setpoints T_(sp) (k) for the next N time steps. Once the optimal temperature setpoints T_(sp) have been determined, the optimal temperature setpoint T_(sp) (k) for the current time step k can be provided to smart thermostat 100 and/or equipment controller 406 and the optimization period can be shifted forward in time by one time step. The temperature setpoint T_(sp) (k) for the current time step k along with feedback received from HVAC equipment 308 and building zone 310 during time step k (e.g., measurements of T_(ia)(k), T_(oa)(k), {dot over (Q)}_(HVAC)(k), etc.) can be used to update the state/disturbance estimates the next time the optimization is performed. Steps 1304-1308 can be repeated at the beginning of each time step to determine the optimal setpoint trajectory T_(sp) for the new (i.e., time shifted) optimization period.

Referring now to FIG. 14, a flowchart of a process 1400 for controlling the temperature of a building zone using model predictive control is shown, according to some embodiments. Process 1400 can be performed by one or more components of model predictive control systems 300-400, as described with reference to FIGS. 3-11. For example, process 1400 can be performed by model predictive controller 302, smart thermostat 100, equipment controller 406, and/or HVAC equipment 308 to control the temperature of building zone 310.

Process 1400 is shown to include identifying parameters of a thermal mass storage model and a HVAC load model for a building zone (step 1402). In some embodiments, step 1202 is performed by system identifier 510. Step 1402 can include generating a thermal mass storage model by modeling the heat transfer characteristics of the building zone using a thermal circuit 800, as described with reference to FIG. 8. Step 1402 may also include generating a HVAC load model by modeling the amount of heating/cooling {dot over (Q)}_(HVAC) provided by HVAC equipment 308 as a function of the temperature setpoint T_(sp) and the zone air temperature T_(ia). The thermal mass storage model and the HVAC load model can be used to generate a system of differential equations that define the relationship between zone air temperature T_(ia), zone mass temperature T_(m), outdoor air temperature T_(oa), the amount of heating or cooling {dot over (Q)}_(HVAC) provided by HVAC equipment 308, the heat load disturbance {dot over (Q)}_(other), and the temperature setpoint T_(sp).

The unknown parameters in the thermal mass storage model and the HVAC load model (e.g., R_(oi), R_(mi), C_(m), C_(ia)) can be organized into parameter matrices A, B, C, and D, as shown in the following equations:

$\begin{bmatrix} {\overset{.}{T}}_{ia} \\ {\overset{.}{T}}_{m} \\ \overset{.}{I} \end{bmatrix} = {{{\begin{bmatrix} {\frac{1}{C_{ia}}\left( {K_{p,j} - \frac{1}{R_{m\; i}} - \frac{1}{R_{oi}}} \right)} & \frac{1}{C_{ia}R_{m\; i}} & {- \frac{K_{p,j}K_{I,j}}{C_{ia}}} \\ \frac{1}{C_{m}R_{m\; i}} & {- \frac{1}{C_{m}R_{m\; i}}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}\begin{bmatrix} T_{ia} \\ T_{m} \\ I \end{bmatrix}} + \mspace{335mu}{\begin{bmatrix} {- \frac{K_{p,j}}{C_{ia}}} & \frac{1}{C_{ia}R_{oi}} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} T_{{sp},j} \\ T_{oa} \end{bmatrix}} + {\begin{bmatrix} \frac{1}{C_{ia}} \\ 0 \\ 0 \end{bmatrix}{{\overset{.}{Q}}_{other}\mspace{79mu}\begin{bmatrix} T_{ia} \\ {\overset{.}{Q}}_{{HVAC},j} \end{bmatrix}}}} = {{\begin{bmatrix} 1 & 0 & 0 \\ {- K_{p,j}} & 0 & K_{I,j} \end{bmatrix}\left\lbrack \begin{matrix} T_{ia} \\ T_{m} \\ I \end{matrix} \right\rbrack} + {\begin{bmatrix} 0 & 0 \\ K_{p,j} & 0 \end{bmatrix}\begin{bmatrix} T_{{sp},j} \\ T_{oa} \end{bmatrix}}}}$ or more compactly: {dot over (x)}=A _(c)(θ)x+B _(c)(θ)u+B _(d) d y=C _(c)(θ)x+D _(c)(θ)u where x^(T)=[T_(ia), T_(m), I], u^(T)=[T_(sp,j), T_(oa)], d={dot over (Q)}_(other)/C_(ia), y^(T)=[T_(ia), {dot over (Q)}_(HVAC,j)], θ is a parameter vector containing all non-zero entries of the system matrices, and

${{A_{c}(\theta)} = \begin{bmatrix} {- {\theta_{5}\left( {\theta_{1} + \theta_{2} + \theta_{4}} \right)}} & {\theta_{5}\theta_{2}} & {\theta_{3}\theta_{4}\theta_{5}} \\ {\theta_{6}\theta_{2}} & {{- \theta_{6}}\theta_{2}} & 0 \\ {- 1} & 0 & 0 \end{bmatrix}},{{B_{c}(\theta)} = \begin{bmatrix} {\theta_{1}\theta_{5}} & {\theta_{4}\theta_{5}} \\ 0 & 0 \\ 1 & 0 \end{bmatrix}},{B_{d} = \begin{bmatrix} \theta_{5} \\ 0 \\ 0 \end{bmatrix}},{{C_{c}(\theta)} = \begin{bmatrix} 1 & 0 & 0 \\ {- \theta_{4}} & 0 & {\theta_{3}\theta_{4}} \end{bmatrix}},{{C_{d}(\theta)} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}},{{D_{c}(\theta)} = \begin{bmatrix} 0 & 0 \\ \theta_{4} & 0 \end{bmatrix}}$

The unknown parameters θ in the thermal mass storage model and the HVAC load model (i.e., the parameters in matrices A, B, C, and D) can be identified by fitting the parameters θ to a set of input-output data (e.g., sets of values T_(ia), T_(oa), T_(sp), and {dot over (Q)}_(HVAC)) collected from the HVAC system when operating in the system identification mode. In some embodiments, the input-output data are collected by modulating the temperature setpoint T_(sp) and observing the corresponding values of T_(ia), T_(oa), and {dot over (Q)}_(HVAC) for a plurality of time steps during the system identification mode. The process of fitting the model parameters θ to the set of input-output data can be accomplished by performing process 900, as described with reference to FIG. 9.

If HVAC equipment 308 are staged, step 1402 can include filtering input-output data to remove the high frequency dither in the indoor air temperature T_(ia) and can compute a time-averaged version of the HVAC load {dot over (Q)}_(HVAC) from the discrete HVAC staging trajectory. The thermal model parameters and HVAC load model parameters θ can then be fit to the input-output data (or filtered input-output data). In some embodiments, step 1402 includes augmenting the resulting state-space model with another integrating disturbance model and estimating the Kalman filter gain for the resulting model. Using data obtained in a secondary experiment or under normal operation, step 1402 can include validating the model through the use of statistics that capture the multi-step prediction accuracy of the resulting model.

Still referring to FIG. 14, process 1400 is shown to include predicting a heat load disturbance {dot over (Q)}_(other) experienced by a building zone at each time step (k=0 . . . N−1) of an optimization period (step 1404). In some embodiments, step 1404 is performed by load/rate predictor 518 and state/disturbance predictor 520. Step 1404 can include generating a disturbance estimate {circumflex over (d)}(k) for each time step k in the optimization period. In some embodiments, step 1404 includes estimating the disturbance state {circumflex over (d)}(k) at each time step k in the optimization period using the following state/disturbance model:

$\begin{bmatrix} {{\hat{T}}_{ia}\left( {{k + 1}❘k} \right)} \\ {{\hat{T}}_{m}\left( {{k + 1}❘k} \right)} \\ {\hat{I}\left( {{k + 1}❘k} \right)} \\ {\hat{d}\left( {{k + 1}❘k} \right)} \end{bmatrix} = {{{A_{aug}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\hat{T}}_{m}\left( {k❘{k - 1}} \right)} \\ {\hat{I}\left( {k❘{k - 1}} \right)} \\ {\hat{d}\left( {k❘{k - 1}} \right)} \end{bmatrix}} + {B_{aug}\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}} + {\begin{bmatrix} K_{x} \\ K_{d} \end{bmatrix}{\left( {\begin{bmatrix} {T_{ia}(k)} \\ {{\overset{.}{Q}}_{HVAC}(k)} \end{bmatrix} - \begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{.}{\hat{Q}}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}} \right)\mspace{79mu}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{.}{\hat{Q}}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}}}} = {{C_{aug}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\hat{T}}_{m}\left( {k❘{k - 1}} \right)} \\ {\hat{I}\left( {k❘{k - 1}} \right)} \\ {\hat{d}\left( {k❘{k - 1}} \right)} \end{bmatrix}} + {D\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}}}$ where

${A_{aug} = \begin{bmatrix} A & B_{d} \\ 0 & I \end{bmatrix}},{B_{aug} = \begin{bmatrix} B \\ 0 \end{bmatrix}},{C_{aug} = \left\lbrack \begin{matrix} C & {\left. 0 \right\rbrack.} \end{matrix} \right.}$ Step 1404 can include recording a history of disturbance estimates {circumflex over (d)}_(hist):={{circumflex over (d)}(k|k−1)}_(k=1) ^(N) ^(hist) predicted by the state/disturbance model.

Step 1404 can include using the history of disturbance estimates {circumflex over (d)}_(hist) to predict the value of {circumflex over ({dot over (Q)})}_(other)(k) at each time step k. In some embodiments, the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) is a function of the time of day, the day type (e.g., weekend, weekday, holiday), the weather forecasts, building occupancy, and/or other factors which can be used by load/rate predictor 518 to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k). For example, step 1404 can include predicting the heat load disturbance using the equation: {circumflex over ({dot over (Q)})}_(other)(k)=f _(pred)(kΔ+t ₀ ,{circumflex over (d)} _(hist)) where Δ>0 is the sample period and t₀ is an initial time. In some embodiments, step 1404 includes using weather forecasts from a weather service 526 and/or load history data from historical data 528 to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) at each time step.

In some embodiments, step 1404 includes using a deterministic plus stochastic model trained from historical load data to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k). Step 1404 can include using any of a variety of prediction methods to predict the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). In some embodiments, step 1404 includes predicting the heat load disturbance {circumflex over ({dot over (Q)})}_(other)(k) using the techniques described in U.S. patent application Ser. No. 14/717,593.

Still referring to FIG. 14, process 1400 is shown to include generating an economic cost function that accounts for a cost of resources consumed by HVAC equipment at each time step (k=0 . . . N−1) of an optimization period (step 1406). An example of an objective function which can be generated in step 1406 is shown in the following equation:

$\min\limits_{T_{sp},\epsilon_{T},{\delta\; T_{sp}}}{\sum\limits_{k = 0}^{N - 1}\left( {{{r_{elec}(k)}{{\hat{\ell}}_{elec}(k)}} + {p_{\epsilon_{T}}{\epsilon_{T}(k)}} + {p_{\delta\; T}\delta\;{T_{sp}(k)}}} \right)}$ where r_(elec)(k) is the price of electricity at time step k, {circumflex over (l)}_(elec)(k) is the predicted electricity consumption of HVAC equipment 308 at time step k, ∈_(T)(k) is a number of degrees by which the zone temperature constraints are violated at time step k, p_(∈) _(T) is a zone air temperature penalty coefficient applied to ∈_(T) (k), δT_(sp) (k) is a number of degrees by which the zone temperature setpoint T_(sp) changes between time step k−1 and time step k, and p_(δT) _(sp) is a temperature setpoint change penalty coefficient applied to δT_(sp)(k). Equation 24 accounts for the cost of electricity consumption (r_(elec)(k){circumflex over (l)}_(elec)(k)), penalizes violations of the indoor air temperature bounds (p_(∈) _(T) ∈_(T)(k)), and penalizes changes in the temperature setpoint (p_(δT)δT_(sp)(k)).

If additional resources (other than electricity) are consumed by HVAC equipment 308, additional terms can be added to the objective function to represent the cost of each resource consumed by HVAC equipment 308. For example, if HVAC equipment 308 consume natural gas and water in addition to electricity, the objective function can be updated to include the terms r_(gas)(k){circumflex over (l)}_(gas)(k) and r_(water)(k){circumflex over (l)}_(water)(k) in the summation, where r_(gas)(k) is the cost per unit of natural gas at time step k, {circumflex over (l)}_(gas)(k) is the predicted natural gas consumption of HVAC equipment 308 at time step k, r_(water) (k) is the cost per unit of water at time step k, and {circumflex over (l)}_(water)(k) is the predicted water consumption of HVAC equipment 308 at time step k.

In some embodiments, step 1406 includes estimating the amount of each resource consumed by HVAC equipment 308 (e.g., electricity, natural gas, water, etc.) as a function of the sensible heating or cooling load {circumflex over ({dot over (Q)})}_(HVAC). In some embodiments, a constant efficiency model is used to compute the resource consumption of HVAC equipment 308 as a fixed multiple heating or cooling load {circumflex over ({dot over (Q)})}_(HVAC) (e.g., {circumflex over (l)}_(elec)(k)=η{circumflex over ({dot over (Q)})}_(HVAC)(k)). In other embodiments, equipment models describing a relationship between resource consumption and heating/cooling production can be used to approximate the resource consumption of the HVAC equipment 308. The equipment models may account for variations in equipment efficiency as a function of load and/or other variables such as outdoor weather conditions (e.g., {circumflex over (l)}_(elec)(k)=f({circumflex over ({dot over (Q)})}_(HVAC)(k), T_(oa)(k), η_(base))). Step 1406 can include converting the estimated heating/cooling load {circumflex over ({dot over (Q)})}_(HVAC)(k) at each time step to one or more resource consumption values (e.g., {circumflex over (l)}_(elec)(k), {circumflex over (l)}_(gas)(k), {circumflex over (l)}_(water)(k), etc.) for inclusion in the objective function.

Still referring to FIG. 14, process 1400 is shown to include estimating initial system states {circumflex over (x)}(0) for a first time step (k=0) of the optimization period (step 1408). The initial system states {circumflex over (x)}(0) may include the estimated zone air temperature {circumflex over (T)}_(ia)(0), the estimated zone mass temperature {circumflex over (T)}_(m)(0), and the estimated integrating disturbance Î(0) at the beginning of the optimization period. In some embodiments, step 1408 is performed by state/disturbance estimator 520. The initial system states {circumflex over (x)}(0) can be estimated using same the state/disturbance estimation model used in step 1404. For example, step 1408 can include estimating the system states {circumflex over (x)}(k) at each time step k using the following equation: {circumflex over (x)}(k+1|k)=A{circumflex over (x)}(k|k−1)+Bu(k)+K(y(k)−ŷ(k|k−1)) ŷ(k|k−1)=C{circumflex over (x)}(k|k−1)+Du(k) where {circumflex over (x)}(k+1|k) is the estimated/predicted state at time step k+1 given the measurement at time step k, ŷ(k|k−1) is the predicted output at time step k given the measurement at time step k−1, and K is the estimator gain.

In some embodiments, the estimated state vector {circumflex over (x)}(k+1|k), the output vector ŷ(k|k−1), and the input vector u(k) are defined as follows:

${{\hat{x}\left( {{k + 1}❘k} \right)} = \begin{bmatrix} {{\hat{T}}_{ia}\left( {{k + 1}❘k} \right)} \\ {{\hat{T}}_{m}\left( {{k + 1}❘k} \right)} \\ {\hat{I}\left( {{k + 1}❘k} \right)} \end{bmatrix}},{{\hat{y}\left( {k❘{k - 1}} \right)} = \begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{.}{\hat{Q}}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}},{{u(k)} = \begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}$ where {circumflex over (T)}_(ia) is an estimate of the zone air temperature T_(ia), {circumflex over (T)}_(m) is an estimate of the zone mass temperature T_(m), Î is an estimate of the integrating disturbance I, {circumflex over ({dot over (Q)})}_(HVAC) is an estimate of the heating or cooling load provided by HVAC equipment 308, T_(sp) is the temperature setpoint, and T_(oa) is the outdoor air temperature. The estimated system states {circumflex over (x)}(0) at the first time step k=0 can be used as the initial system states.

Still referring to FIG. 14, process 1400 is shown to include using a thermal mass storage predictive model to constrain system states {circumflex over (x)}(k+1|k) at time step k+1 to the system states {circumflex over (x)}(k) and the temperature setpoint T_(sp)(k) at time step k (step 1410). An example of a constraint which can be generated based on the thermal mass storage predictive model is:

$\begin{bmatrix} {{\hat{T}}_{ia}\left( {k + 1} \right)} \\ {{\hat{T}}_{m}\left( {k + 1} \right)} \\ {\hat{I}\left( {k + 1} \right)} \end{bmatrix} = {{A\begin{bmatrix} {{\hat{T}}_{ia}(k)} \\ {{\hat{T}}_{m}(k)} \\ {\hat{I}(k)} \end{bmatrix}} + {B\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}} + {B_{d}{{\overset{\overset{\bigwedge}{.}}{Q}}_{other}(k)}}}$ where {circumflex over (T)}_(ia)(k+1) is the predicted temperature of the building zone at time step k+1,{circumflex over (T)}_(m)(k+1) is the predicted temperature of the building mass at time step k+1, Î(k+1) is the predicted value of the integrating disturbance at time step k+1, {circumflex over (T)}_(ia)(k) is the predicted temperature of the building zone at time step k, {circumflex over (T)}_(m)(k) is the predicted temperature of the building mass at time step k, Î(k) is the predicted value of the integrating disturbance at time step k, T_(sp) (k) is the temperature setpoint at time step k, T_(oa)(k) is the outdoor air temperature (measured) at time step k, and {dot over ({circumflex over (Q)})}_(other)(k) is the estimated heat load disturbance at time step k.

Process 1400 is shown to include using a HVAC load predictive model to constrain the HVAC equipment load {dot over ({circumflex over (Q)})}_(HVAC)(k) at time step k to systems states {circumflex over (x)}(k) and the temperature setpoint T_(sp)(k) at time step k (step 1412). An example of a constraint which can be generated based on the HVAC load model is:

$\left\lbrack {{\overset{\overset{\bigwedge}{.}}{Q}}_{other}(k)} \right\rbrack = {{C\begin{bmatrix} {{\hat{T}}_{ia}(k)} \\ {{\hat{T}}_{m}(k)} \\ {\hat{I}(k)} \end{bmatrix}} + {D\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}}}$ where {dot over ({circumflex over (Q)})}_(HVAC)(k) is the predicted HVAC equipment load at time step k and {circumflex over (T)}_(ia)(k), {circumflex over (T)}_(m)(k), Î(k), T_(sp)(k), and T_(oa)(k) are the same as the corresponding variables in the thermal mass storage model.

Still referring to FIG. 14, process 1400 is shown to include optimizing the economic cost function subject to constraints defined by the predictive models and constraints on the predicted zone temperature {circumflex over (T)}_(ia)(k) to determine optimal temperature setpoints T_(sp) for the building zone for each time step (k=0 . . . N−1) of the optimization period (step 1414). The objective function can be optimized subject to a set of constraints (e.g., Equations 25-34). The constraints may include the thermal mass storage model and the HVAC load model, which define the relationship between {circumflex over ({dot over (Q)})}_(HVAC), the temperature setpoints T_(sp), and the zone air temperature T_(ia) at each time step. The constraints may also include constraints on the zone air temperature T_(ia)(k) and constraints that define the penalty terms in the objective function. The optimization performed in step 1414 may generate a temperature setpoint T_(sp) (k) for each time step in the optimization period.

Process 1400 is shown to include providing the temperature setpoint T_(sp) for the first time step (k=0) of the optimization period to an equipment controller (step 1416). Step 1416 can include sending the optimal temperature setpoints T_(sp) from model predictive controller 302 to an equipment controller 406. In some embodiments, both model predictive controller 302 and equipment controller 406 are components of a smart thermostat 100 (as shown in FIG. 4). In other embodiments, model predictive controller 302 is separate from smart thermostat 100 and configured to send the optimal temperature setpoints to smart thermostat 100 via a communications network 304 (as shown in FIG. 3). In some embodiments, only the temperature setpoint for the first time step (i.e., T_(sp)(0)) is provided to the equipment controller in step 1416. The remaining temperature setpoints are for future time steps and may be updated the next time the optimization is performed (e.g., at the beginning of the next time step).

In some embodiments, step 1416 includes providing the entire set of temperature setpoints T_(sp) (i.e., a temperature setpoint T_(sp) (k) for each time step k) to the equipment controller. In the event that communication between model predictive controller 302 and equipment controller 406 is lost (e.g., network connectivity is disrupted), equipment controller 406 can use the set of temperature setpoints provided in step 1416 until communications between model predictive controller 302 and equipment controller 406 are restored. For example, equipment controller 406 can use each temperature setpoint T_(sp)(k) received in step 1416 to control HVAC equipment 308 during the corresponding time step k until communications between model predictive controller 302 and equipment controller 406 are restored.

Still referring to FIG. 14, process 1400 is shown to include using feedback collected from the HVAC system during the first time step (k=0) to update the thermal mass storage predictive model (step 1418). The feedback collected from the HVAC system may include measurements of the zone air temperature T_(ia) and/or measurements of the HVAC equipment load {dot over (Q)}_(HVAC). Step 1418 may include calculating the difference between the value of the zone air temperature {circumflex over (T)}_(ia)(k|k−1) predicted in step 1408 and the actual measured value of the zone air temperature T_(ia)(k) collected in step 1418. Similarly, step 1418 may include calculating the difference between the value of the HVAC equipment load {circumflex over ({dot over (Q)})}_(HVAC)(k|k−1) predicted in step 1408 and the actual measured value of the HVAC equipment load {dot over (Q)}_(HVAC)(k) collected in step 1418.

The updating performed in step 1418 may include multiplying the Kalman gain matrix K generated in step 1402 by the vector of differences, as shown in the following equation:

$\begin{bmatrix} {{\hat{T}}_{ia}\left( {{k + 1}❘k} \right)} \\ {{\hat{T}}_{m}\left( {{k + 1}❘k} \right)} \\ {\hat{I}\left( {{k + 1}❘k} \right)} \\ {\hat{d}\left( {{k + 1}❘k} \right)} \end{bmatrix} = {{A_{aug}\begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\hat{T}}_{m}\left( {k❘{k - 1}} \right)} \\ {\hat{I}\left( {k❘{k - 1}} \right)} \\ {\hat{d}\left( {k❘{k - 1}} \right)} \end{bmatrix}} + {B_{aug}\begin{bmatrix} {T_{sp}(k)} \\ {T_{oa}(k)} \end{bmatrix}} + {\begin{bmatrix} K_{x} \\ K_{d} \end{bmatrix}\left( {\begin{bmatrix} {T_{ia}(k)} \\ {{\overset{.}{Q}}_{HVAC}(k)} \end{bmatrix} - \begin{bmatrix} {{\hat{T}}_{ia}\left( {k❘{k - 1}} \right)} \\ {{\overset{.}{\hat{Q}}}_{HVAC}\left( {k❘{k - 1}} \right)} \end{bmatrix}} \right)}}$ In some embodiments, the updating performed in step 1418 is equivalent to the updating step of a Kalman filter in which the predicted states {circumflex over (x)}(k+1|k) are calculated based on the error between the predicted system output ŷ(k|k−1) at time step k and the actual/measured values of the predicted output variables y(k).

Process 1400 is shown to include shifting the optimization forward in time by one time step (step 1420) and returning to step 1408. Steps 1408-1420 can be repeated at the beginning of each time step to generate a set of temperature setpoints T_(sp) for each time step in the shifted optimization period. The first temperature setpoint T_(sp) (0) generated for the shifted optimization period can be provided to the equipment controller and used to control the HVAC equipment.

Performance Graphs

Referring now to FIGS. 15-18, several graphs 1500-1800 illustrating the performance of model predictive control (MPC) systems 300-400 are shown, according to some embodiments. Graph 1500 compares the energy consumption of MPC systems 300-400 to the energy consumption of a baseline temperature control system without MPC. Graph 1600 indicates the cost of electricity (i.e., $/kWh) which varies over time. Line 1502 represents the energy consumption of the baseline system, whereas line 1504 represents the energy consumption of MPC systems 300-400. MPC systems 300-400 use more energy than the baseline system during an off-peak period 1506 when energy prices are low. However, MPC systems 300-400 use less energy than the baseline system during a peak period 1508 when energy prices are high. Advantageously, this results in a cost savings relative to the baseline system.

Graph 1700 compares the building mass temperature trajectories of the baseline system and MPC systems 300-400. Line 1702 represents the temperature T_(m) of the solid mass within building zone 310 when the baseline system is used to provide temperature control. Since the baseline system does not store energy in the building mass, the temperature T_(m) remains relatively constant over the duration of the optimization period. Line 1704 represents the temperature T_(m) of the solid mass within building zone 310 when MPC systems 300-400 are used to provide temperature control. During the off-peak period 1506, building zone 310 is precooled, which results in a decrease in the temperature T_(m) of the building mass. During the peak period 1508, thermal energy from the air within building zone 310 is moved into the building mass, which increases the temperature T_(m) of the building mass and provides cooling for the air within building zone 310.

Graph 1800 compares the zone air temperature trajectories of the baseline system and MPC systems 300-400. Line 1802 represents the temperature T_(ia) of the air within building zone 310 when the baseline system is used to provide temperature control. The temperature T_(ia) remains relatively constant over the duration of the optimization period. Line 1704 represents the zone air temperature T_(ia) within building zone 310 when MPC systems 300-400 are used to provide temperature control. During the off-peak period 1506, building zone 310 is precooled to the minimum comfortable zone temperature T_(min) (e.g., 70° F.). The zone air temperature T_(ia) remains relatively constant during the off-peak period 1506, but the temperature of the building mass T_(m) decreases as heat is removed from building zone 310. During the peak period 1508, the building zone temperature T_(ia) is allowed to increase to the maximum comfortable zone temperature T_(max) (e.g., 76° F.). Heat from the air within building zone 310 flows into the building mass, which provides cooling for the air within building zone 310 and increases the temperature of the building mass T_(m).

System Architecture

Referring now to FIG. 19, a block diagram of a system 1900 is shown, according to an exemplary embodiment. The system 1900 provides for multiple smart thermostats 100 (and corresponding sets of HVAC equipment 308) to be enabled with model predictive control functionality via communication with a shared model predictive control server 1902. The system 1900 of FIG. 19 is shown to include multiple sets of HVAC equipment 308, multiple smart thermostats 100, a provider computing system 1904, multiple user mobile devices 1906, a utility provider computing system 1908, and a weather service 1910. The provider computing system 1904 is shown to include a model predictive controller 1902, a security and authorization system 1912, a model predictive control (MPC) application programming interface (API) 1914 and a thermostat API 1916.

As shown in FIG. 19, the system 1900 includes multiple smart thermostats 100. Each smart thermostat 100 may be associated with (e.g., installed at) a different space, for example a different house, commercial building, building zone (e.g., different apartment or office in a shared building), etc. In the example shown, each smart thermostat 100 is communicable with and configured to control a set of HVAC equipment 308 which serves the space associated with the smart thermostat 100. For example, in some embodiments each smart thermostat 100 is located in and associated with a separate house and is configured to control the HVAC equipment 308 (home furnace, air conditioning system, etc.) for that house. Each smart thermostat 100 may be associated with a different end user (e.g., owners of the various houses). In various implementations, the system 1900 may thereby serve many (e.g., several, dozens, hundreds, thousands, etc.) spaces (e.g., homes, apartments, condos, buildings, offices, retail spaces, etc.) which, in some cases, are owned, leased, and/or occupied by many different people.

The system 1900 is also shown to include a provider computing system 1904. The provider computing system 1904 may be implemented using servers, cloud-based computing resources, and/or other computing platforms. The provider computing system 1904 may be associated with a provider of smart thermostat services. The provider computing system 1904 is communicable with the smart thermostats 100 via a network (e.g., network 304 as described with reference to FIG. 3). The provider computing system 1904 is also shown as being communicable with multiple user mobile devices 1906, a utility provider computing system 1908, and a weather service 1910, for example via the network 304.

The thermostat API 1916 facilitates interoperability of the smart thermostats 100 with the provider computing system 1904 (e.g., the MPC API 1914 and the security and authorization system 1912), the user mobile devices 1906, and, in some embodiments, the utility provider computing system 1912. The MPC API 1914 facilitates interoperability of the model predictive controller 1902 with other elements of the provider computing system 1904 (e.g., the security and authorization system 1912 and the thermostat API 1916), the weather service 1910, and the utility provider computing system 1912.

The security and authorization system 1912 is configured to manage access of the multiple smart thermostats 10 and user mobile devices 1906 to the features provided by the provider computing system 1904. The security and authorization system 1912 is configured to provide a tokenized authentication process. For example, it may be the case that only a subset of the smart thermostats 100 are authorized (e.g., due to subscription models, regulatory compliance issues, etc.) to interoperate with the model predictive controller 1902. In such an example, the security and authorization system 1912 may receive a user name and password combination from a user mobile device 1906 and/or from a smart thermostat 100, and may use the user name and password combination to verify authorization of that user and the corresponding smart thermostat 100 to access the features of the model predictive controller 1902. As another example, the security and authorization system 1912 may be configured to facilitate secure (e.g., encrypted and authenticated) communications between the smart thermostats 101 and the model predictive controller 1902 by applying a data authentication process and/or encryption/decryption process at the MPC API 1914 and thermostat API 1916.

The weather service 1910 is configured to provide weather reports and/or forecasts to the MPC API 1914. The weather service 1910 may be a governmental or non-governmental meteorological system capable of providing weather data corresponding to the location(s) of the smart thermostats 100. In some cases, the smart thermostats 100, the HVAC equipment 308, and the spaces served thereby may be located at geographically dispersed locations such that the weather is different at the various spaces. In such cases, the weather data provided by the weather service 1910 may include a set of data corresponding to each geographic location associated with a smart thermostat 100. The geographic locations of the smart thermostats 100 may be automatically determined, for example using IP addresses associated with the smart thermostats 100, or input by a user via a smart thermostat 100 or a user mobile device 1906. Multiple weather services 1910 may be used to provide sufficient geographic coverage.

The system 1900 may include one or more utility provider computing systems 1912 associated with the one or more utility providers that may provide utility resources (e.g., electricity, natural gas, etc.) consumed by the HVAC equipment 308. The utility provider computing system(s) 1912 are configured to provide utility rate information to the provider computing system 1904 (e.g., to the MPC AIP 1914 and/or the thermostat API 1916). Utility rates may be time-varying, for example such that the price of a unit of a utility resource (e.g., one kilowatt) may change each hour, each day, every 15 minutes, etc. The utility provider computing systems 1912 may provide the current rate and/or historical or future rates.

The utility provider computing system(s) 1912 may also provide signals relating to demand response programs, for example incentive-based demand response programs. In one example herein, a utility provider may provide incentives (e.g., monetary credit, redeemable rewards points, prizes, etc.) for customers who agree to reduce energy consumption during peak periods defined by the utility provider, for example by lowering a temperature setpoint by a predefined amount over the peak period or otherwise loosening constraints on inside air temperature over the peak period. In such an example, a utility provider computing system 1912 may be configured transmit an indication of an upcoming peak period to the provider computing system 1904 to allow the provider computing system 1904 to control one or more smart thermostat(s) 100 to participate in demand response program as described in detail below. The MPC API 1914 and/or the thermostat 1916 may be configured to provide data back to the utility provider computing system(s) 1912, for example indicating whether or not a particular smart thermostat 100 participated in the demand response program for a particular peak period. The system architecture of FIG. 19 provides for various such electronic communications to facilitate participation in various demand response programs.

The model predictive controller 1902 is configured as described above for the model predictive controller 302, adapted to provide model predictive control functionality for multiple smart thermostats 100. For example, the model predictive controller 1902 may be configured to execute a separate parameter identification (system identification) process for each building or space served by a smart thermostat 100 enrolled in an MPC mode. The model predictive controller 1902 may thereby obtain, for each smart thermostat 100, a predictive model of the thermal dynamics of the space served by the corresponding HVAC equipment 308. The model predictive controller 1902 may then use each predictive model to generate temperature setpoints for the corresponding smart thermostat 100 as described in detail above.

In some embodiments, the model predictive controller 1902 includes a database in addition to the various elements shown for model predictive controller 302 in FIG. 5. The database may be configured to store, for each smart thermostat 100 served by the model predictive controller, profile information for the smart thermostat 100 (e.g., a network address or other fully-qualified reference, a geographic location, an associated utility provider, etc.), temperature setpoint schedules or ranges, and identified parameters for a thermal model for the space corresponding to the smart thermostat. In some embodiments, the database of the model predictive controller 1902 stores historical data (e.g., loads, weather, rates) for each smart thermostat 100.

The model predictive controller 1902 may be configured to pull data from the database and use that data in a model predictive control process, for example as described with reference above to the model predictive controller 302. In various embodiments, the model predictive controller 1902 is configured to perform model predictive control processes for multiple smart thermostats 100 in series or in parallel. In either case, providing model predictive control functionality for many smart thermostats 100 via unified remote computing resource (i.e., via the model predictive controller 1902) may reduce the total amount of computational resources required to provide such functionality, as compared to implementing the model predictive controller 302 separately for all spaces and smart thermostats 100. Furthermore, complex computations may be shifted to a computing resource remote form the smart thermostat 100 and HVAC equipment 308, thereby reducing the hardware requirements at the smart thermostat 100.

The architecture of system 1900 may also facilitate updates and improvements to the model predictive control process over time via software updates to the model predictive controller 1902. Additionally, the architecture of system 1900 may facilitate learning based on aggregation of thermal models for many spaces. For example, in some embodiments, the model predictive controller 1902 can use the multiple models to improve the efficiency of model training/identification processes, for example by using a model from a similar space to provide initial guesses for parameters of the predictive model for a new space.

Training Data Collection for Smart Thermostats

Referring now to FIGS. 20-24, a process for collecting building data for use identifying predictive models for a use in model predictive control is illustrated. FIG. 20 shows a flowchart of a process 2000 for obtaining training data, according to an exemplary embodiment. FIGS. 21-24 illustrate user interfaces that may be presented on a smart thermostat 100 to facilitate process 2000, according to exemplary embodiments. Process 2000 may be executed by the system 1900 of FIG. 19, the system 300 of FIG. 3, or the system 400 of FIG. 4 in various embodiments, and the present disclosure contemplates various distributions of the functions of process 2000 across the computing resources of the system 1900 of FIG. 19, the system 300 of FIG. 3, or the system 400 of FIG. 4.

At step 2002 of process 2000, a user is prompted to select an option to enable a model predictive control (MPC) mode. For example, the smart thermostat 100 may display a graphical user interface that includes a button, toggle, etc. selectable by a user to enable a model predictive control mode. FIG. 21 shows an example of such a user interface, according to some embodiments. As shown in FIG. 21, the display screen 102 of the smart thermostat 100 displays text 2102 explaining an energy aid (MPC) mode to a user and a toggle 2100. The toggle 2100 is selectable by a user to enable (or disable) an MPC mode (e.g., by touching the toggle 2100 on the display screen 102). In some embodiments, the user may be prompted to select an option to enable the MPC mode via a user mobile device 1906.

In response to a user selecting the option to enable the MPC mode (e.g., via the smart thermostat 100, at step 2004 a determination is made regarding whether sufficient historical training data are available for model identification. For example, in some embodiments, building and weather data (e.g., indoor and outdoor air temperatures, temperature setpoints, equipment operating points, etc.) are collected and stored during normal operation of the smart thermostat 100. For example, the data may be stored within a memory device of the smart thermostat 100, at a model predictive controller 302, at a model predictive controller 1906, or in some other computer memory resource. To determine whether the historical training data is sufficient at step 2004, in some embodiments the amount of available historical training data is assessed by determining whether a number of time steps or a duration of time represented in the historical training data exceeds a threshold value. In other embodiments, to determine whether the historical training data is sufficient at step 2004, a system identification process is executed and the prediction error of the identified model is compared to a maximum acceptable error (e.g., a one-step ahead or multi-step ahead prediction error), such that the training data is considered insufficient if the resulting model does not generate predictions within the acceptable error.

If the sufficient historical training data are determined to be available at step 2004, process 2000 proceeds to step 2016 wherein a system model is identified using the historical training data, for example as described in detail above. If sufficient historical training data are determined to not be available, process 2000 proceeds to step 2006 where the user is prompted for permission to initiate a system identification experiment. FIG. 22 shows an example of a user interface on the display device 102 of the smart thermostat 100 the may be provided at step 2006. As shown in FIG. 22, text 2200 provides information regarding the system identification experiment, for example to inform the user that a test is needed to allow energy and cost savings to be optimized and that the user will lose control of the temperature setpoint during the test. The display 102 is also shown as providing a button 2202 selectable by the user to choose to start the system identification experiment at a present time. In some embodiments, the smart thermostat 100 can provide the user with an option to schedule the system identification experiment for a future time.

In response to a user selection to initiate a system identification experiment, the process 2000 proceeds to step 2008, step 2010, and step 2012, which may be executed substantially simultaneously. At step 2008, the system identification experiment is executed by varying temperature setpoints over the duration of a system identification experiment to induce a dynamic response of indoor air temperature. For example, setpoints may be varied over a duration of the system identification experiment using an algorithm designed to increase the amount of robust training data acquired over the time period, for example as described in U.S. patent application Ser. No. 16/516,928 filed Jun. 19, 2019, the entire disclosure of which is incorporated by reference herein. In some embodiments, the duration of the system identification experiment is twenty-four hours (i.e., one day), allowing for capture of system dynamics across a full day. In other embodiments, a different duration is used (e.g., twelve hours, two days, one week, one month, etc.).

At step 2010, during execution of step 2008, an indication is provided to the user that the system identification experiment is ongoing. FIG. 23 shows an example user interface that can be displayed at step 2010, in which a test indicator 2300 is provided on the display 102 of the smart thermostat 100 to inform a user that the system identification experiment is ongoing. As shown in FIG. 23, the display 102 also shows a current indoor air temperature 2302, a heating setpoint 2304, and a cooling setpoint 2306. The heating setpoint 2304 and/or the cooling setpoint 2306 may show the current setpoint being applied by the system identification experiment executed at step 2008. For example, as shown in FIG. 23, a cooling setpoint of seventy degrees is less than an indoor air temperature of seventy two degrees, such that the setpoint used by the system identification experiment is causing excitation of system dynamics that may drive the indoor air temperature from seventy two degrees towards seventy degrees. By displaying information regarding the ongoing system identification experiment via the display 102, an occupant of a space is allowed to easily see and understand the cause of any temperature fluctuations and/or discomfort caused by the system identification while verifying that the smart thermostat 100 and the HVAC equipment 308 are operating properly.

At step 2012, a user is prevented from modifying temperature setpoints during execution of the system identification experiment of step 2008. Preventing user modification of setpoints allows for the system identification experiment to be executed consistently and in an automated manner configured to generate robust training data without interruptions caused by user adjustments. FIG. 24 shows a message 2400 that may be provided on the display 102 of the smart thermostat 100 in response to a user's attempt to modify a temperature setpoint during execution of a system identification experiment. The message 2400 informs the user that the system identification experiment is ongoing and that a purpose of the system identification experiment is to enable the smart thermostat 100 to provide energy and cost savings for the user. The message 2400 also includes a timer 2402 that indicates when the system identification experiment will end, after which point the user may be allowed to modify setpoints for the smart thermostat 100. Preventing a user from overriding or prematurely terminating the system identification experiment may improve the reliability of process 2000 and reduce the amount of incomplete/insufficient data stored by the system.

At step 2014, training data is collected from execution of the system identification experiment (i.e., the execution of step 2008). For example, temperature setpoints, indoor air temperature, weather data, equipment setpoints, etc. may obtained and saved at the model predictive controller 1902. At step 2016, a system model is identified based on the training data, for example as described above with reference to FIG. 9. In some embodiments, step 2016 is executed using approaches described in U.S. patent application Ser. No. 15/963,324 filed Apr. 13, 2018, U.S. patent application Ser. No. 15/900,459, filed Feb. 20, 2018, U.S. patent application Ser. No. 16/447,724, filed Jun. 20, 2019, and/or U.S. patent application Ser. No. 16/418,715, filed May 21, 2019, all of which are incorporated by reference herein in their entireties. The HVAC equipment 308 can then be controlled based on the identified system model as described in detail above.

Model Predictive Controller with Demand Response

Referring now to FIG. 25-30, flowcharts and example user interfaces illustrating a demand response mode for the smart thermostat 100 are shown, according to exemplary embodiments. FIGS. 25-30 contemplate a demand response program in which a utility provider offers incentives (e.g., monetary incentives, discounts/credit on utility bills, redeemable rewards points, prizes, etc.) to customers who agree to allow their thermostats to be automatically adjusted during utility-provider-designated peak hours. For example, in some implementations the utility provider may provide an incentive when a cooling setpoint is increased by a designated amount (e.g., four degrees) during a peak period (e.g., mid-afternoon on a hot summer day). In other examples, the utility provider may provide an incentive if some other energy-savings mode, comfort range, etc. is applied by a thermostat during the designated peak hours. With reference to the architecture of FIG. 19, these types of demand response programs are made possible by real-time or ahead-of-time communication from a utility provider computing system 1912 that indicates that the current time is being treated as a peak period or specifying a future time period that will be treated as a peak period.

FIG. 25 shows a flowchart of a process 2500 for enrolling a thermostat in a demand response program, while FIGS. 26-27 show user interfaces on the display 102 of the smart thermostat 100 that can be used in executing process 2500. Process 2500 may be executed by the system 1900 of FIG. 19, the system 300 of FIG. 3, or the system 400 of FIG. 4 in various embodiments, and the present disclosure contemplates various distributions of the functions of process 2500 across the computing resources of the system 1900 of FIG. 19, the system 300 of FIG. 3, or the system 400 of FIG. 4.

At step 2502, a particular utility provider associated with a particular smart thermostat 100 is identified. As described with reference to FIG. 19, a provider computing system 1904 may serve many smart thermostats 100 that may be located in different geographic regions and, accordingly, may control HVAC equipment 308 that consumes utility resources supplied by a set of multiple utility providers. Step 2502 may include selecting the utility provider that is relevant for a particular smart thermostat 100 from the set of multiple utility providers. For example, the provider computing system 1904 (e.g., the security and authorization system 1912) may store a mapping between geographic locations (e.g., zip codes, GPS coordinates, city/state/country names, etc.) and the utility providers that serve those geographic regions. In this example, step 2502 includes determining the geographic location of the smart thermostat 308 (and/or the HVAC equipment 308), for example based on user input of a geographic location to a smart thermostat 100 or a user mobile device 1906, GPS data collected by a smart thermostat 100 or a user mobile device 1906, or based on an IP address associated with a smart thermostat 100 or a user mobile device 1906. In most instances, a utility provider can be selected based on the geographic location (e.g., due to a monopoly held by the utility provider). In other embodiments, a user is prompted (e.g., via the smart thermostat 100, via a user mobile device 1906) to directly select a utility company from a list of utility companies (e.g., an alphabetical list, a text-searchable list).

At step 2504, a determination is made regarding whether a demand response program is available from the relevant utility provider. For example, a look-up table of utility providers and associated incentive programs may be used to find the available programs based on the utility provider identified at step 2504. In some embodiments, a query is sent to a utility provider computing system 1912 to prompt the utility provider computing system 1912 for confirmation of whether a demand response program is available for a particular smart thermostat 100.

If a demand response program is not available from the relevant utility provider (e.g., due to a policy or pricing scheme of the utility provider), at step 2506 an indication is provided to a user that a demand response program is not currently available. For example, a message stating that the user's utility provider does not offer a demand response program may be displayed on the display 102 of the smart thermostat 100.

If a demand response program is determined to be available for the relevant utility provider at step 2504, at step 2508 the user is prompted to opt-in to peak hours savings, i.e., to enroll/participate in the demand response program. For example, FIG. 26 shows a selectable toggle 2600 presented on the display 102 of the smart thermostat 100 that can be selected to enable (or disable) a peak savings mode. The user interface of FIG. 26 also shows fields 2602 that provide information regarding temperature bounds that will be used during peak hours.

In response to receiving a user selection of the option to opt-in to peak hours savings, at step 2510 peak savings temperature bounds are obtained which will be applicable during peak hours. For example, as shown in FIG. 27, a user interface may be presented on the display 102 of the smart thermostat 100 that allows a user to input a comfortable cooling (or heating) range for a space served by the HVAC equipment 308 controlled by the smart thermostat 100. As shown in FIG. 27, a temperature selection widget 2700 allows a user to scroll through available numbers (e.g., integers) to select a lower bound and an upper bound on a comfort range. The view in FIG. 27 may be reached by selecting field 2602 in FIG. 26.

In other embodiments, step 2510 includes obtaining the peak savings temperature bounds by automatically assigning the peak savings temperature bounds based on a policy of the utility provider. For example, in some embodiments, the terms of the demand response program may require use of a utility-provider-defined set of peak savings temperature bounds in exchange for the incentive offered by the demand response program. As another example, the utility provider may require at least a minimum difference between the temperature setpoint/bounds used during normal operation and peak savings temperature bounds used for participation in the demand response program. In such an example, the peak savings temperature bounds for a particular smart thermostat 100 can be calculated using the differential required by the utility provider and temperature bounds from a normal operating mode or schedule. The temperature bounds obtained at step 2510 may be stored by the model predictive controller 1902 or the mode predictive controller 302.

A smart thermostat 100 may thereby be enrolled in a demand response program. The smart thermostat 100 can then be controlled to participate in the demand response program, for example as shown in FIG. 28. FIG. 28 shows a flowchart of a process 2800 for participating in a demand response program is shown, according to an exemplary embodiment. FIGS. 29-30 show example user interfaces that may be shown by the display 102 of the smart thermostat 100 as part of process 2800. Process 2800 may be executed by the system 1900 of FIG. 19, the system 300 of FIG. 3, or the system 400 of FIG. 4 in various embodiments, and the present disclosure contemplates various distributions of the functions of process 2800 across the computing resources of the system 1900 of FIG. 19, the system 300 of FIG. 3, or the system 400 of FIG. 4.

At step 2802, an indication is received from the utility provider of a peak period. In the example of FIG. 19, a utility provider computing system 1912 provides an indication (e.g., message, signal, electronic communication, etc.) of a peak period to the MPC API 1914 and/or the thermostat API 1916. The indication may include details of the timing of the peak period (e.g., start time, end time, duration). In some cases, the peak period indicated by the utility provider computing system 1912 may begin at approximately the same time the indication is transmitted to the provider computing system 1904. In other cases, the indication my provide information of an upcoming peak period, for example provided one or more hours before the peak period.

At step 2804, an opportunity is provided for the user of a smart thermostat 100 to opt-out of participating in the demand response program for the particular peak period (i.e., for the peak period indicated in step 2802). FIG. 29 shows an example of a user interface that may be presented on the display 102 of the smart thermostat 100 to allow the user to opt-out (or opt-in) to participating in the demand response program. In the example shown, the display 102 shows text 2900 describing the peak period (e.g., start and end times) and explaining that the thermostat 100 will be controlled to participate in the demand response program for that peak period if the user does not opt-out. The display 102 also provides an opt-out button 2902 that can be selected to opt-out of the identified peak period, and a continue button 2904 that can be selected to opt-in to the identified peak period. In the embodiment shown, the smart thermostat 100 participates in the demand response program for the peak period if no response from the user is received (e.g., before the beginning of the peak period, after a threshold amount of time). In other embodiments, affirmative opt-in is required for participation in the demand response program for a particular peak period.

In some embodiments, two optimizations are performed over the peak period, including a first optimization executed using a normal temperature setpoint or comfort range and a second optimization executed using a peak saving temperature bounds. The optimizations may be performed as described in detail above. As a result of the optimizations, two predicted costs of operating the HVAC equipment over the peak period are generated, including a predicted cost of operating the HVAC equipment subject to the normal temperature setpoint/bounds and a predicted cost of operating the HVAC equipment subject to the peak saving temperature bounds. Step 2804 may include comparing the two costs and displaying a predicted savings with the prompt to opt-out of peak savings for the particular peak period. The predicted savings can help a user determine whether to opt-out of the peak savings mode for the particular peak period.

Step 2806 asks whether an opt-out is received. If an opt-out is received from the user (e.g., if the opt-out button 2902 is selected by a user), then process 2800 proceeds to step 2808 where the HVAC equipment 308 is control as per a normal operational mode. For example, a temperature setpoint or comfort range may remain as defined by a normal operating schedule. In some cases, a model predictive control process operates as described above based on the normal schedule to provide temperature setpoints to the smart thermostat 100. Accordingly, at step 2808, the demand response program is not participated in for the peak period identified at step 2802. In some embodiments, the smart thermostat 100 provides a message via the thermostat API to the utility provider computing system 1912 indicating that smart thermostat 100 is not participating in the demand response program for the identified peak period.

If the user does not opt-out of the demand response program for the peak period (or an affirmative opt-in is received, for example by selection of continue button 2904), the process 2800 proceeds to step 2810. At step 2810, constraints for the model predictive control process over the peak period are defined using peak savings temperature bounds, for example as obtained in process 2500. For example, the values of T_(max), T_(min) and/or T_(sp,min), T_(sp,max) in equations 28-30 above may be defined over the peak period using the peak savings temperature bounds. The model predictive controller 1902 or model predictive controller 302 can thereby update the constraints on the optimization of the objective function to transition from a normal operating mode (in which a first set of values of T_(max), T_(min) and/or T_(sp,min), T_(sp,max) is used) and the peak savings mode to participate in the demand response program (in which a second set of values of T_(max), T_(min) and/or T_(sp,min), T_(sp,max) is used).

As noted above, the values of T_(max)(k), T_(min)(k) and/or T_(sp,min)(k), T_(sp,max)(k) may be time-variant. Accordingly, the constraints may be defined over a time horizon that includes at least a portion of a peak period in addition to time steps outside of a peak period. In a situation where advanced notice of a peak period is obtained (i.e., the indication of step 2802 identifies a future time step as the start of the peak period), each of T_(max)(k), T_(min)(k) and/or T_(sp,min)(k), T_(sp,max)(k) may be defined to have different values before k=s and after k=s, where k=s indicates the starting time step of the peak period. As one example, taking an illustrative scenario where the peak temperature bounds are offset from the normal temperature bounds by six degrees, the minimum temperature setpoint may be defined as

${T_{{sp},\min}(k)} = \left\{ \begin{matrix} {{70{^\circ}\mspace{14mu}{F.}},} & {{k = 1},\ldots\mspace{14mu},{s - 1}} \\ {{64{^\circ}\mspace{14mu}{F.}},} & {{k = s},\ldots\mspace{14mu},h} \end{matrix} \right.$ while the maximum temperature setpoint may be defined as

${T_{{sp},\max}(k)} = \left\{ {\begin{matrix} {{74{^\circ}\mspace{14mu}{F.}},} & {{k = 1},\ldots\mspace{14mu},{s - 1}} \\ {{80{^\circ}\mspace{14mu}{F.}},} & {{k = s},\ldots\mspace{14mu},h} \end{matrix}.} \right.$ The constraints can then be used to optimize over a time horizon that includes time steps both within the peak period and outside the peak period (e.g., before the peak period, after the peak period).

At step 2812, an optimization is performed subject to the constraints defined at step 2810 to determine temperature setpoints for the smart thermostat over a time horizon that includes at least a portion of a peak period. The optimization and other details of a model predictive control process are described in detail above with reference to at least FIGS. 12-14. The optimization may be performed over a horizon that includes time steps both inside and outside the peak period, in which case step 2812 facilitates transition from a normal operating mode to the peak savings operating mode. Step 2812 can be performed multiple times (e.g., as explained with reference to step 1420) as the peak period elapses to generate temperature setpoints for the smart thermostat during the peak period. Step 2812 may be performed by the model predictive controller 1902 or the model predictive controller 302.

At step 2814, equipment (e.g., HVAC equipment 308) is controlled in accordance with the temperature setpoints generated as a result of the optimization of step 2812. In the architecture of system 1900 of FIG. 19, the model predictive controller 1902 may perform an optimization subject to the constraints (at step 2812) and provide the resulting temperature setpoints to the corresponding smart thermostat 100 via the MPC API 1914 and/or the thermostat API 1916. The smart thermostat 100 may then control the HVAC equipment 308 associated with the smart thermostat 100 to drive the indoor air temperature at the space to the temperature setpoint identified by the model predictive controller 1902. For example, equipment commands (e.g., on/off decisions, operating capacities, damper/valve positions, fan speeds) can be generated by smart thermostat 100, for example using a proportional-integral (PI) controller, a proportional-integral-derivative (PID) controller, or some other approach.

The HVAC equipment 308 is thereby controlled to participate in the demand response program for a peak period. In some cases, communications between the smart thermostat 100 and the utility provider computing system 1912 are used to confirm that the smart thermostat 100 participated in the demand response program for the peak period and to confirm that the utility provider computing system 1912 recorded the participation and that the incentive will be awarded to the user. In some embodiments, a notification (e.g., via the smart thermostat 100, via a user mobile device 1906) is provided to a user congratulating the user on successful participation in the demand response program.

In some embodiments, for example as shown in FIG. 30, the smart thermostat 100 may provide a user with an option to opt-out of an ongoing peak period (i.e., to cancel participation in the demand response program for a current peak period). As shown in FIG. 30, the display 102 of the smart thermostat 100 includes an opt-out button 3000 that a user can select to escape participation. In response to a user selection of opt-out button 3000, the temperature constraints may be reset to those corresponding to a normal schedule and the optimization may be rerun. The user may not receive the incentive associated with the demand response program.

Energy Savings Analytics and Visualization

Referring now to FIG. 31, a graphical user interface 3100 is shown, according to an exemplary embodiment. The graphical user interface 3100 may displayed on a user mobile device 1906 and/or the display 102 of a smart thermostat 100. The data visualized on the graphical user interface 3100 may be generated by the model predictive controller 1902 or the model predictive controller 302.

The graphical user interface 3100 is shown to include a graph 3102 of energy usage versus time. For example, the graph 3102 shows hours used (e.g., total time for which HVAC equipment is on) for each day in the last week. Other units of energy consumption, power consumption, equipment utilization, etc. may be used in other embodiments, as may other time scales. The graph 3102 includes a standard usage line 3104 that represents the amount of energy that would have been consumed by the equipment if controlled under a standard approach, for example if controlled according to a predefined setpoint or schedule. The model predictive controller may be configured to generate the data used to plot the standard usage line 3104. For example, the predictive thermal models described above may be used with the standard predefined setpoint or schedule to estimate energy consumption over the plotted time period.

The graph 3102 also includes an actual usage line 3106 that shows the actual energy consumption of the equipment, as controlled by the model predictive controller 1902 or model predictive controller 302. The actual usage line 3106 may represent measurements (e.g., energy meter data) or estimates, for example estimates based on control signals (e.g., equipment commands) or optimal temperature setpoints generated by the optimization processes described herein. Icons 3108 may indicate energy-usage-related events, for example peak periods for which the smart thermostat 100 participated in a demand response program.

By showing the actual usage line 3106 the standard usage line 3104 together on the graph 3102, a user can easily see the benefits provided by the model predictive control processes and system architectures described herein. User trust in the autonomous optimization processes described herein may thereby be substantially increased, increasing user adoption of the technological advantages described herein.

As can be seen from the graph 3102, substantial reductions in energy usage by the equipment 308 can be achieved using the systems and methods described herein, both on the individual level (e.g., for individual homeowners, residents, or building managers) and in aggregate across many users of many smart thermostats 100 as in FIG. 19. Energy costs savings, improved sustainability, reduced carbon emissions, reduced peak demands on energy grids, etc. are all technical advantages of the present disclosure.

Configuration of Exemplary Embodiments

The construction and arrangement of the systems and methods as shown in the various exemplary embodiments are illustrative only. Although only a few embodiments have been described in detail in this disclosure, many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.). For example, the position of elements can be reversed or otherwise varied and the nature or number of discrete elements or positions can be altered or varied. Accordingly, all such modifications are intended to be included within the scope of the present disclosure. The order or sequence of any process or method steps can be varied or re-sequenced according to alternative embodiments. Other substitutions, modifications, changes, and omissions can be made in the design, operating conditions and arrangement of the exemplary embodiments without departing from the scope of the present disclosure.

The present disclosure contemplates methods, systems and program products on any machine-readable media for accomplishing various operations. The embodiments of the present disclosure can be implemented using existing computer processors, or by a special purpose computer processor for an appropriate system, incorporated for this or another purpose, or by a hardwired system. Embodiments within the scope of the present disclosure include program products comprising machine-readable media for carrying or having machine-executable instructions or data structures stored thereon. Such machine-readable media can be any available media that can be accessed by a general purpose or special purpose computer or other machine with a processor. By way of example, such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to carry or store desired program code in the form of machine-executable instructions or data structures and which can be accessed by a general purpose or special purpose computer or other machine with a processor. Combinations of the above are also included within the scope of machine-readable media. Machine-executable instructions include, for example, instructions and data which cause a general purpose computer, special purpose computer, or special purpose processing machines to perform a certain function or group of functions.

Although the figures show a specific order of method steps, the order of the steps may differ from what is depicted. Also two or more steps can be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. Likewise, software implementations could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various connection steps, processing steps, comparison steps and decision steps. 

What is claimed is:
 1. A system comprising: a plurality of thermostats corresponding to a plurality of HVAC systems that serve a plurality of spaces; a computing system communicable with the plurality of thermostats via a network, the computing system configured to, for each space of the plurality of spaces, obtain a set of training data relating to thermal behavior of the space; identify a model of thermal behavior of the space based on the set of training data; perform a model predictive control process using the model of thermal behavior of the space to obtain a temperature setpoint for the space; and provide the temperature setpoint to a thermostat corresponding to an HVAC system serving the space; wherein the plurality of thermostats are configured to control the plurality of HVAC systems in accordance with temperature setpoints corresponding to each.
 2. The system of claim 1, wherein the computing system is configured to obtain the set of training data relating to thermal behavior of the space by executing a system identification experiment for the space, wherein executing the system identification experiment comprises providing a series of temperature setpoint changes to the thermostat corresponding to the HVAC system serving the space.
 3. The system of claim 2, wherein the thermostats are configured to prompt users for permission to perform the system identification experiment, and wherein executing the system identification experiment comprises causing the thermostat to display an indication that the system identification experiment is ongoing.
 4. The system of claim 2, wherein executing the system identification experiment comprises preventing a user from adjusting a current temperature setpoint for the space.
 5. The system of claim 1, wherein the computing system is configured to identify the model of thermal behavior of the space by performing a system identification process to determine values for parameters of the model of thermal behavior.
 6. The system of claim 1, wherein the computing system is configured to receive data from a utility provider computing system and adjust the model predictive control process based on the data from the utility provider computing system. 